In a previous section I learned that the Russian word for shoes was обувь. So what's туфли?
О́бувь is used for footwear in general, while ту́фли refers to a specific kind of footwear.
Anything from this:
Any shoe in the traditional sense. So no sneakers, boots, slippers, etc. It is the general shoe that one would wear to job or school or any occasion where things like sneakers and running shoes aren't acceptable.
I believe yes (however, my English is not good enough and I'm not really sure I understand the meaning of 'formal shoes' correctly).
Sports footwear is not included in «ту́фли», neither are boots that have a top of the boot (голени́ще), neither is anything made of rubber.
Somewhere in the short/long lessons the picture of sneakers was connected to the word ту́фли. Should I report that the next time I see it?
Yes, sneakers are definitely not туфли. Туфли normally don't include sports footwear.
I think sneakers usually correspond to кроссо́вки, but I think the English 'sneakers' is broader than the Russian 'кроссо́вки'.
What word is used in Russia to refer to "flip flops"? (When I was a kid, my parents called them "shower shoes"; I've also heard them referred to as thongs, although nowadays that more often means something else.) And by the way, are flip flops popular in Russia, like they are here in America?
"Flip flops" has several names in Russian: "шлё́пки", "шлё́панцы" (the first two words refer to a word "шлё́пать" - slap, smack, pat, ie description of the sound while walking), "сла́нцы", "вьетна́мки". It is very popular in summer in the South of Russia.
For me, боти́нки usually refers to something higher, like this:
There is an overlap in the meaning of ту́фли and боти́нки, so for many items of footwear, both words can be used. I personally use the word ту́фли much often.