i/y is a hard thing to deal with - Czech children spend at school several years of Czech classes just to master the skill when to write which. So no big deal when you make mistake there, less educated people in CZ also mistake those.
However a handy thing to remember:
There is always i after these letters: š č ř ž j c
There is always y after h ch k r
Only exceptions are foreign words as "kino" or "cyklus"
Similarly, American children struggle with spelling in words containing 'e' following/preceding 'i.' And we have a little rhyme: "'i' before 'e' except after 'c', or when pronounced like an 'a' as in 'neighbor' and 'weigh.'
I wonder if there a Czech mnemonic for the i/y dilemma?
We remember "hychykyrydytyny", "žščřcjďťň" and "befelemepeseveze".
For the last group we then learn the list of words that use y, as the others use i. We call the list ("vyjmenovaná slova" "specified words") https://cs.wikipedia.org/wiki/Vyjmenovan%C3%A1_slova http://prirucka.ujc.cas.cz/ ).
I want to talk about pronunciation. In my reading of the lightbulb lessons, and other sources, I think the 'y' in 'myši' and the 'y' in 'kočky' should be pronounced the same. Is that correct? - sorta like 'i' in English 'bit'? And these should also be the same as the 'i' in 'ti'? To my American ears, 'kočky' sounds more like 'kočkee,' while 'myši' sounds more like I think it's supposed to, less 'beet' more 'bit.'
Do you mean the length or the narowness? In "bit" the wovel is wider and longer than "beet" at the same time.
Czech has only one short i-like vowel and one long. Any wider/narrower differences are not phonemic and will be perceived as the same vowel by Czech listeners.
The actual phonetic realization can be slightly variable between different persons and regions.
In "myši a kočky", both 'y's as well as the 'i' are pronounced /ɪ/ - the same as in English "bit". All short 'y's and 'i's in Czech are pronounced like that. And all long 'ý's and 'í's are pronounced /i:/ - the same as in English 'beet'. For English speakers, these two vowels are one of the easiest bits of Czech phonology.