First, you have to realize that descendants of Muhammad =/= Shi'ite. In fact, just based on statistics, they are more likely to be Sunni than Shi'ite. You'll find descendants in both groups though.

Supposedly, people with the last name "Syed" or "Hasani" or "Husaini" or "Shareef" are descendants of Muhammad. I say supposedly because realistically speaking, the vast majority of these claims are simply not verifiable and obviously, there is quite an incentive to claim descent.

With that said, there are some people whose descent is pretty certain. The current king of Jordan is one of them. During WW1, Sharif (<--- note the title) Husain, a definite descendant of Muhammad, rebelled against the Ottomans in exchange for the British giving three of his sons kingship. One became king of Arabia (before losing it to the current monarchy), the other became king of Iraq (that didn't last very long), and the third became king of Transjordan and his descendants still hold the throne of Jordan.

You also have to keep in mind that while theologically, Shi'ites revere the family of Muhammad, reality is a bit more nuanced. The Shi'ite population of Iraq was not at all happy with Sharif Husain's son becoming king of Iraq. Even historically, one of the earliest branch points of Shi'ism is the split of the Zaydis and Ithna 'Asharis. Zayd was a descendant of Muhammad but his refusal to distance himself from Abu Bakr and 'Umar led to a large portion of the Shi'ite community deserting him (and eventually leading to a second Karbala event). As far as I know, technically speaking, both groups of Shi'ite still revere him to this day but as you can see, reverence does not necessarily lead to accepting as a leader.

Yes. The majority of Shiites no longer have living imams--the largest group, Twelvers, who dominate in Iran, Iraq, Bahrain, Azerbaijan, and are a substantial minority in Lebanon, Kuwait, Saudi Arabia, Afghanistan (the Hazara), as well as a smaller minority in South Asia and beyond, recognize Twelve Imams, with the last one "occluding himself" in the 9th or 10th century (depending on if you're counting from the "minor" or "major" occultation). So they haven't had a real world descendent of Mohammed through his son-in-law Ali since that point. The next biggest group is probably the Zaydis (Fivers) who live in Yemen (if you've heard of the Houthis, they're Zaydi) and have a different theological relationship with their imams (who are not seen as as infallible). They also had religious leadership based on descent from Mohammed until at least the 1960's, but I am not sure about religious leadership since then. They also switched between different branches of the lineage, and there were multiple periods were their were rival claimants. Then there are groups like the Turkish and Kurdish Alevis and more Orthodox Arab Alawites who follow a similar schema in terms of Imams to the Twelvers, but these groups have other differences from Orthodox Twelvers. Finally, there are still existing groups with active lineages living imams--the biggest being the Nizari Ismailis led by the Aga Khan and various Bohra groups (the biggest being the Dawoodi Bohra). The current Aga Khan, by their count, is the 49th Imam and the current rival heads of the Dawoodi Bohra, called the Da'i al-Mutlaq (there's a succession dispute since the last Da'i died in 2014), both claim to be the 53rd Imam. These chains are unbroken (though there are obviously a lot of splits when two rival both claimed the title) so there's not really a need for separate documents.

It should be said that in many places Sunnis also give special treatment or at least respect to descendents of Mohammed. For example, the King of Jordan claims to be the descendent of the Mohammed as does the current King of Morrocco. Some of these ties are authentic, some are more historically questionable, and here it's harder to adjudicate the claims because there are long periods where these families were not prominent. Many of the claims are documented back centuries, however. Even in the Sunni world, lineage ties to Mohammed's family were used to cement legitimacy. Generally, these people are called Sayyids or Sharifs.

The kings of Jordan and Morocco, and the Ismaili Aga Khans are some prominent people who claim descent from Muhammad. Direct patrilineal descendants of Ali and Fatima, in general, are called Sayyids.

A study that looked at Indian Sayyids didn't find any evidence of common patrilineal ancestry (if there are real Sayyids in that group, their lineage isn't modal, so they don't stand out). AFAIK, no other studies have been done in other countries. It would require much more data to determine who is or isn't a Sayyid (or if they even exist).