Read this
That’s almost 300 posts of a Medicare professional explaining all of this stuff.
In your question number 1 you start talking about Part D, but end asking about Part C. Is that a typo or are you confused? Part C & D are mutually exclusive; you can’t have both.
In a nutshell, Part C (Medicare replacement) replaces all the other stuff with just part C which you get only from a private insurer, but Medicare pays most of the premiums. If you get part C you can’t get (and don’'t need) anything else. part C works OK if your broke but they make their money through massive care rationing. So don’t get an expensive disease and you’ll be fine.
The other alternative, if you can afford it is …
If you want Part A & B (Traditional medicare) you order those from Medicare’s website.
It would be foolhardy not to get Part D as well, which you get only from private insurance companies. If you don’t get part D you will go broke paying full retail for meds as you get older.
If you go with parts A, B, & D it would be foolhardly not to also get some sort of Medicare Supplement (“medicap”) plan. If you don’t you will go broke paying out of pocket for all the stuff traditional Medicare doesn’t cover, or only covers partly. Medicare itself (A & B) is much much less complete than any modern employer-provided health insurance you may be familiar with, and the Medicare Supplement plans fill in those many shortfalls.
You get those Supplement plans from private insurers. There are many gradations of Supplement plans, all labeled by letter. “Part G” being the most comprehensive plan.
Yes, dental and eye care is totally separate from all the above. Buy it or not as you prefer.
If you have not gotten the Medicare book from Medicare you need to get it and read it.
There are mistakes you can easily make that will cost you thousands of dollars for the rest of your life with no ability to have a do-over. Read that thread and the book. Then come back with detailed questions.
Lots of us have done this recently and can provide good advice once you understand the basic outline of how the alphabet soup works.