In Polish, if you want to say "This cat!" (notice the exclamation mark), you say "Ten kot!". On the other hand, "This is a cat." would be "To jest kot." The difference is that "ten" indicates the male gender and is very specific about something. "To" is neuter and indicates that something rather unknown is about to be described. The sentence would be perfectly fine if you left the "jest" out, "To jest kot." and "To kot." are both correct and have the same meaning, "to" does not replace "jest".
Also look at the following sentence: "Ten kot to kot mojego dziadka." (This cat is my grandfather's cat.) It first describes a very specific cat ("Ten kot ...") and later describes it ("... to kot mojego dziadka.") Also note that "To kot mojego dziadka." would rather be used if you saw a single cat, as opposed to "Ten kot to kot mojego dziadka" if you saw at least two cats.
The construction with "to" is an archaism which evolved from a special type of sentence structure in Slavic languages, which was used to emphasize the subject of some statement. Schematically, it can be described as: "(det) Subj... Pron (referring to Subj) Verb-Phrase", e.g. "Einstein... he was a smart man!", "The dog... it is man's best friend." or "The money... don't think about it!" In Polish it seems that the copula (i.e. jest) has become redundant in this type of sentences and hence is omitted, however, in other Slavic languages it is still compulsory (for instance in Bulgarian). So in conclusion - you can use "to" if it makes sense in English to say "A, he/she/it is B".
"E" sounds like "E" from "end" or "epic" or "especially"
I don't think that you have an english equivalent of "Y" sound, sadly. I can't think of any words...
I suggest you to put some words with "Y" in google translate and play the sound. Like "łyżwy", "mały", "syty". So you can find the difference by yourself.
Mostly: masculine ones end in a consonant, femine ones in -a, and neuter in -o or -e. But the number of exceptions is not that small. Still, this should be your first presumption.
Some clearly masculine words like "mężczyzna" and "tata" end in -a, but by definition, they have to be masculine as they describe men.
You mixed some stuff. Firstly, the notion of being 'animate' is only important for masculine nouns in Accusative. So for example "Widzę małego kota" (I see a small cat) vs "Widzę mały stół" (I see a small table).
Secondly, if you have a sentence like "X is Y" and Y is an adjective, it just stays in Nominative. And the alternative (when Y is a noun phrase) is Instrumental (Ten kot jest małym zwierzęciem = This cat is a small animal), not Accusative.
The adjective (przymiotnik) has to agree with the noun:
masculine singular noun -> This old house - Ten stary dom
feminine singular noun -> This old book - Ta stara książka
neuter singular noun -> This old tree - To stare drzewo
masculine plural noun -> These old houses - Te stare domy
feminine plural noun -> These old books - Te stare książki
neuter plural noun -> These old trees - Te stare drzewa