/u/Iphikrates can certainly answer your questions better than I can, but the Greeks most definitely knew about boots. There are, according to the New Pauly, 82 words known in Greek for different kinds of footwear, many of which are boots. Many of these types of footwear are named after the location or people where they are common. For example, we have Περσικαί (Persian shoes) and Λακωνικαί (Spartan shoes), which Aristophanes joked as being quintessentially manly. Others are named after the people who invented them--amusingly enough (at least if /u/Iphikrates actually does show up) Diodorus mentions a kind of light army boot called after him "iphicratids" (ἰφικρατίδες). Others were just called...well, what they were called. The most common form of footwear in Greece (unlike at Rome) was the sandal, which came in quite a lot of forms and for which there are a lot of names. But sandals were not really working gear, and the New Pauly provides a nice little chart of four of our 82 different kinds of shoe (tell me if you can't see that I'll try to upload it somewhere elseI've changed the link, it should work now). Of the four shoes listed there the ἐνδρομίς is a sort of thin boot associated with races and hunters (Artemis wore these kinds of shoes, and the name ἐνδρομίς is derived from δρόμος, "a race"). The κόθορνος (or "buskin" as it is translated sometimes) was a sort of high boot laced up the front and coming up to the mid-calf or so--they were associated with tragedy and grandeur and were either open-toed or had an upturned curly toe like you see in the picture. The κρηπίς was a sort of half-boot or ankle boot--according to Lewis and Short these were often associated with soldiers and martial virtue. Finally the ἐμβάς was a sort of felt shoe for ordinary walking--the word is related to the verb ἐμβαίνω, "to step, walk."

As for "authentic" Greek sandals...I've never seen any, and the one's you've got there look like ordinary designer sandals with ridiculous names and outrageous prices ($200 for a pair of flip flops, are you nuts!). There were a lot of different kinds of sandals but Greek sandals were generally relatively simple. One very important distinction between pretty much everything I've seen and reality is that Greeks laced their sandals up, as they did their shoes--the complex leather ribbons that criss-cross the foot diagonally to hold the shoe in place that you see on so many girls' "ancient" sandals were a development of the Roman Period. Buckles were apparently known (at least by the Roman Period), but Greek sandals were generally laced up the ankle--see this guy lacing up his sandals (he also seems to have some sort of legging or sock underneath? I dunno) or this guy in the middle (I think those might actually be technically κρηπῖδες, I'm not sure. Same basic idea though).

Regarding question #3, hoplite reenactment is definitely a thing, so even if few retailers sell "Greek-style" sandals ,though they do here, there are definitely hobbyists you could find who do make them. I'd recommend you start with Hoplites.org.

Edit: As I'm not sure that responding to your only non-historical question alone is allowed, you should listen to u/xenophontheathenian, quality answer.