Explain the origin of the word "tomcat", please. I well know word "cat", but at first time see word "tomcat". What is it exactly mean? What is the different?
"Tomcat" is what this course uses to signify that the cat is male.
In reality it's very uncommon to refer to a male cat as a tomcat, you would usually just call it a cat.
It is not very common in this course either ;) We translate it as a "tom" a few times just for fun, but mostly use "cat" and allow both кот and кошка when translating back to Russian.
The only problem here is that the word itself is very common, so pretending кот did not exist was not a good option. We still do not have дача and колбаса, which are nigh on untranslatable without really long footnotes.
I googled "Дача" and indeed it does not seem so easy to translate (I would venture "country house" as a first approximation), but Wikipedia, Wiktionary and Google Translate all seem to imply that "колбаса" is simply "sausage", without any caveat. Is there some hidden trick here?
Yes, exactly. It is the partitive use of Genitive, expressing an indeterminate amount of something (or an amount only determined by how much was actually used, consumed, bought etc.).
Genitive is a very popular case which comes in handy in many places and with many prepositions.
Would it be plain wrong to use the accusative or would it be correct but change the meaning slightly and/or sound odd? In French you could say Le chat veut du lait = the cat wants some milk = partitive but you could also say Le chat veut le lait = the cats wants the milk -> it sounds a bit strange but it could be said , it just changes the focus from the amount to the product itself
Both can be used. The difference is that shade of meaning between "wanted the milk, not something else" and "wanted to have some milk".