Both are correct, actually. It can be a shorter form of "Ciebie", but here it is plural of "ten" = "these".
Fun fact: when writing something directly to another person (a letter/a text message/a facebook message/etc.) it's polite to capitalize all forms of 'you': Ci, Tobie, Twój, etc. It is not done in this course because these are just sentences without any context. But because of that, sometimes some people mistakenly capitalize 'ci' meaning 'these', although it's in fact a different word :D
Do I read ci like ći, ć, or ci? :D Also dzieci - is it dzieć, dzieći or dziec-i? Do I ALWAYS read ci as ć?
Sometimes we simplify and say that 'this is the same sound', but yeah, if it's 'ci' there is some 'i' sound. So maybe 'dzieći' seems the closest? Sometimes it really matters, like "koń" (singular Nominative) and "koni" (plural Genitive).
"mężczyźni" is the basic, Nominative form of "men", "mężczyznami" is Instrumental. Apart from other usages that you will learn, a noun phrase needs to be in Instrumental after a form of "być" (to be).
No, it is a part of the subject of the sentence (These men), so there's no reason for it to use any other case than old, plain Nominative.
Accusative forms of neuter nouns are same as their nominative forms, in both singular and plural.
I wouldn't say that, but in a way...
"ci" = "these" and "tamci" = "those".
And then you have to remember different concepts of 'closeness': Polish will use ci/ci/tamci, while English will use these/those/those.