Is polish a regular language?
Regular language, I mean, not with many exeptions and weird things going on?
It is, and it isn't.
Find an obscure word in Polish dictionary which I've never seen before and tell me its basic definition and as a native I'll most likely be able to inflect it correctly. But there is so many rules for all of that, that it can't be easilly memorized. Some of them are never taught at schools despite being subconciously applied in normal speech.
Probably best thing you can do is to just go with the flow and learn from your mistakes.
Since you speak a lot more languages than I do, I'm not sure if you'd find Polish as extraordinarily difficult as I do. But in my opinion, there really are a lot of weird grammar things going on in Polish. It may well be a regular language (as some Polish commenters have claimed), but it certainly has a bewildering number of rules. E.g. some nouns have 2 plural forms, e.g. if you're speaking about years, then it's 1 rok, 2-4 lata, 5-21 lat, 22-24 lata, 25-31 lat, 32-34 lata... Usually, in a positive sentence, the object takes the accusative, but if there's a negation, then it takes the genitive. And in the genitive, male nouns either take the ending -a or -u, but there doesn't seem to be a rule for that - or at least there is such a number of rules that they might as well be non-existent. Then they don't only have 3 different (but hellishly similar-looking) sets of rules for the declension of masculine, feminine and neuter nouns (in 6 cases x 2 for singular and plural), there's another set of rules for the declension of masculine-alive nouns (like boy, farmer, dog) in 7 cases, plus fruit and cigarettes are also considered to be alive. I find it an incredibly entertaining language to learn (I've been learning Polish for about 2 years now), but I have long since given up on the idea that I will ever be able to spontaneously come up with a correct Polish sentence of more than 2 or 3 words.
You are almost correct, except that it's not genitive singular, but nominative plural. You were right about genitive plural part though.
PS: As some may have noticed, "rok" (year) and "lato" (summer) shares the same plural form – "lata". We would literally count years in summers instead. :) And in more poetic language you can count somebody's age in springs (wiosny).
Animate (alive) and personal masculine nouns in genitive always take -a and inanimate (almost) always take -u. The harder thing is to distinguish if a noun is animate or not. Especially if it is a cigarette. But it is still somehow regular as for three syllables ending in -os are usually animate. But it really is not a rule to remember I think. It's more to get used to when listening and speaking to just feel what you should say. Polish is not a language where you should memorize every rule.