1) How many years have had multiple days on which more than one F5 tornado occurred?
Officially, never
Unofficially, 2011. In addition to the obvious (4/27), 5/24 very obviously produced 3 and possibly 4 (Canton Lake) EF5 level tornadoes.
Also unofficially: 1965 - Palm Sunday, of course, but 5/8/1965 saw both the official F5 in Colome, SD as well as a - per sources anyway - probable F5 near Wolbach, NE.
Here's a local news video talking about the 50th anniversary of the Wolbach-Primose tornado
And those 2 years are it.
2) How many times in tornadic history have F5 or EF5 tornadoes occurred on back-to-back days?
Officially, never.
Unofficially, there are 3 strong possibilities:
June 8-9th, 1953 (Flint, MI followed by Worcester)
April 27th-28th, 2014 (Vilonia followed by Louisville, MS)
and possibly: May 19-20th, 2013 (Shawnee followed by Moore)
Here's a post from locomusic01 a while ago about Shawnee. Talk about an overshadowed tornado.
https://talkweather.com/threads/significant-tornado-events.1276/post-77461