In regards to your first question, the reasons fall into two groups: for the good of the object, and to facilitate learning from that object.

1. Museums are the best place for them for the same reason it's better to hire an electrician than to wire your house yourself. You might do it right, but do you really want to take the risk of setting your home on fire if you don't need to? Collections care & management is a complex job and it's incredibly easy to mess things up. Once you do? You might be able to minimize the appearance of damage, but you'll never be able to undo it.

I'm going to be veering into natural history with my examples because that's my area of expertise, but a collections manager who works with artifacts would come up with a comparable list.

There are a lot of things you have to learn to be a collections manager. The breadth of topics you need to consider in order to keep things preserved as long as possible and restore/repair them when needed is astounding. And the more material types you have to work with, the more complex it gets.

What's the climate like in your collections room? Average, minimum, and maximum temperatures? How rapidly does it fluctuate? Same for relative humidity. What's the optimum temperature and relative humidity for the materials making this specific object? What will keep them from warping or cracking? Which minerals and fossils will experience a chemical reaction that slowly eats them away and is nigh-impossible to stop (and expensive to even try) if their environment is too humid or it fluctuates too much? Which ones will disappear if exposed to air? What's the safe range if you don't have enough funding to have good climate control in your collections?

What about safety? Did you know that some old photographs can "spontaneously" combust? Do you know how to keep that from happening? Which minerals are hazardous and how do you create a boundary between you and them? Was that old taxidermied animal preserved with arsenic? If you don't have firsthand knowledge, do you know how to tell?

Do you know which minerals you can't store next to each other because sometimes they interact and one comes out looking worse for the wear? Are your cabinets and drawers earthquake-secured if you live in a tectonically active area? Can you shell out for those pricey fireproof cabinets? How's the glass on your display cabinet? Did you think about making sure it's treated to prevent both UVA and UVB from entering and making your pigments fade or plastics become brittle?

What kind of containers and labels are you using? Are they acid-free so they don't corrode your objects? Is your orpiment inside the right type of plastic container that will prevent leaching? Did you remember to keep its label outside its bag so the label doesn't get eaten over time? Speaking of labels, have you kept yours appropriately labelled? Included all the information, right? Doesn't matter how good your memory is, right it all down immediately* because I promise you, you will forget something important about it eventually.

How's your pest control? You don't have cockroaches, do you? 'Cause they'll eat labels. So will silverfish. And your objects themselves if they're made of old paper. Hope you don't get moths eating those priceless old garments.

I could go on, but I'll stop there 'cause if you've managed to stay focused, your head's probably spinning. But check out this website, which is a great resource for even experience museum professionals. The "Resources" tab is particularly relevant here.

EDITED TO ADD: If you have your own personal collection, please do check out this site! I hate to see people's faces fall when they realize their personal treasures have fallen afoul of easily avoidable degradation. A couple quick tips: The biggest source of object damage is physical damage due to handler error. So be gentle, use drawers that have auto-stops on them (or pull the drawer beneath out halfway before opening the drawer you're interested in to catch it in case it falls), and learn how to build supports for very fragile objects. If your precious objects include books and you experience a flood, the first thing you need to do is freeze your books. That will halt a very fast and unfixable damage process while you triage the rest. They can stay frozen while you fix everything else. When you're trying to decide where to store things in your home but don't have space in your general living area, a general rule of thumb is that attics are awful (huge temperature and humidity fluctuations) and basements are better (more stable environment, like a cave) so long as things are elevated off the floor in case it floods. And most things should be kept away from windows since they introduce both environmental fluctuations and light. Cool, dark, and stable is the way to go.

2. You're an avid collector and have a huge collection? Awesome, passionate people are amazing fonts of knowledge and resources! ....You did keep a detailed database so its searchable, right? (The answer to that is almost always no, but oh so wonderful when it's yes). Are you able to set up a website so people can search it to do research? Complete with helpful pictures and full keystroke data? How likely is someone to know about your collection? Oh, you don't want visitors to come study your important one-of-a-kind fossil? But we know it's a new species and you have the only one we know about. Oh, you don't care. Okay, then. Or, alternatively, you're happy to have visiting researchers, but it's going to be harder to coordinate visits with you than it would be to coordinate with a collections manager who's always there during work hours because it's their job. And you probably don't have a good photography station, do you?

Part two, unlike part one, has a bit of controversy surrounding it. On the one hand, what's the point of preserving something if it's not used? Is it better to keep something in a drawer at a museum where it isn't being accessed by researchers? Or is it better that it belong to someone who gets enjoyment out of having it? There are camps of people who say everything belongs in a museum and camps who say nothing does, but most lie on the continuum between. I personally ask myself, "How rare is it? How likely is it to provide new information? Do we already have a large enough sample in dedicated institutes?"...questions like that. It's easier to answer the first and third questions, though still not always straightforward. "Rare" is a relative metric. What's considered a "good sample size" has changed over time. The second question is hard to peg down because we might think something has given us all the data it can, but then something like genetic testing gets invented and suddenly we're finding new species in collections that were misidentified as something else because they're cryptic. There's a balance that needs to be struck between possession in public institutes and possession by private individuals. Everyone has a different idea of where the fulcrum lies.

There's also the problem of low funding for museums, and mismanagement when politicians and businesspersons occupying the administrative seats decide to not listen to the museum professionals working beneath them. If the life is being strangled out of museum, many of their objects are going to be sitting in dusty old corners, not even looked at for years. In an ideal world, that would not be the case, but it's an unfortunate fact right now. There are pushes being made to bring those specimens to the public's attention in new ways, such as those that make use of digital tools not available in the past (particularly social media), but it's still a situation of not enough workers, not enough money, and too much to do.

EDITED to add some arguments for the existence of private collections, since I forgot to include them: Amateur enthusiasts are a source for many amazing posthumous collection donations to public institutes. Every collection like that is one that the museum wouldn't have gotten otherwise (particularly if it was collected at great personal expense, then donated to a museum with little to no funding for acquisitions). Even if a person doesn't have an amazing collection, a small middling-quality one can still spark a lifelong love for a subject that they go on to infect others with (which is part of the point of a museum to begin with).

So, ONI_Spook has already given a thorough answer laying out why museums are good places from the perspective of preservation and education.

However, the display of antiquities and artifacts in museums also has an ethical dimension. There are several categories of situations where it is not appropriate for objects to be held and displayed in museums.

Probably the foremost category would be the display of human remains. In the 19th century and into the 20th century, it was common for museums to display the remains of native americans as well as goods taken from grave sites. In the same vein, the remains of Sarah Baartman, known as 'the hottentot venus' were displayed in a museum in Paris for a century and a half.

However, mores have changed, and now there is a much greater degree of cultural sensitivity concerning the handling of human remains and grave goods. It is widely understood that it is not appropriate to put on display the bones of the ancestors of living Native American peoples. For example, in the United States, the National Museum of the American Indian has a repatriation policy for the return of human remains, funerary goods and objects of cultural patrimony to the living descendants.

A second problematic category is antiquities or artifacts which have been looted. Again, mores surrounding this have changed over the decades. Many many objects gathered by European antiquarians and amateur archaeologists in the 19th century were taken out of their original context in the Middle East, Asia, Africa or the Americas, and brought to museums and private collections in Europe and the United States. An example of this would be Priam's Treasure a collection of artefacts taken by Heinrich Schliemann from the site of Troy in anatolia, and brought to Germany in the 1880s. Or, the Elgin Marbles which were removed from Ottoman-ruled Athens between 1800-1805 and brought to Britain, and are now housed in the British museum.

There are also cases of outright looting, such as the Benin bronzes which were looted by british soldiers during the conquest of the Benin kingdom, and housed in the British museum and private collections. Or the taking of Qing statues during the Opium wars, and ended up in private art collections in France.

Again, attitudes of museums, arts and antiquities collectors, and international laws have changed. There now exist strong international conventions and national laws that make dealing in looted art illegal, and strengthening the consensus that artifacts should normally remain in their countries of origin.

As you may have noticed if you clicked through the links, all the objects I mentioned have been the object of international controversy concerning their repatriation. The government of Greece has repeatedly demanded the return of the Elgin Marbles to Athens, which the British museum has thus far refused. Ditto, Nigeria has demanded the return of the Benin bronzes as objects looted during war. Again British museum has refused, arguing that if they had to return objects collected in dubious circumstances during the 19th century, their collections would be much diminished. The Qing bronzes have actually been returned to China, after some controversy. Fights over the control of antiquities of outstanding cultural value can sometimes get imbued with the issue of national prestige for the source countries.

So, the provenance of an object does matter, and museums strive to only accession objects which have not been looted or otherwise been put on the black market.

Of course, there are also arrangements where objects can be loaned or placed in temporary exhibits. For instance, the National Museum of Egypt is the permanent home of the Tutankhamen treasures, but they periodically travel overseas for special exhibits.

So, it is ironic that you mention Indiana Jones. Both the Ark of the Covenant and the Hovito golden idol would now be considered looted artifacts, and their display in a museum today in the United States would most likely result in legal and diplomatic challenges from Egypt and Peru seeking their repatriation.

EDIT- I'd also recommend this article by Patrick Hunt as further reading on the ethics and laws regarding antiquities trafficking.