Translation:I am busy until the weekend.
Till is spelt with two ells, it's not short for until, either. However, till was spelt til centuries ago, though it's still not a shortening; until is derived from it, in same way that unto relates to to, and that's why until only has one ell (it formed before the spelling change).
They are two different verbs. Shimasu is conjugated from suru (する) which means to do. Imasu is conjugated from iru (いる), which means to be/to exist (only for animated objects like humans and animals. Inanimate objects demand aru) ねこ が います。it is a dog. いす が ありまあす。it is a chair.
It can be really confusing in the beginning! Just keep reading about verbs and grammatical constructions, and you'll get there :) I use a dictionary app called takoboto to look up words I am unsure about :)
i'm sorry, but i'm pretty sure that there is something wrong with both of your example sentences. "Neko ga imasu" translates to "A cat exists" or "There is a cat".
The misunderstanding comes from the double meaning of the english verb "to be". On the one hand it means something along the lines of "is equal to"/" is part of". Take the sentence "Duo is a bird", where "is" works essentially as an equal sign (=). When used like this, the verb "to be" is grammatically called "copula"; the copula construct can be found in many languages, including German and Spanish (which even has two copula verbs). The copula meaning of "to be" translates to "desu", so "Duo wa tori desu" -> "Duo = bird" -> "Duo is a bird".
The other meaning of "to be" is when it refers to the actual existence of something. The sentence "Duo is" means "Duo exists". In this case the translation of "to be" would be "imasu" for animate subjects (people and animals) and "arimasu" for inanimate subjects (plants, furniture, other things). So it goes like this: "Duo ga imasu" -> "Duo" is something that exists -> "Duo is" or "There is a Duo".
The way I go about it is to always check whether I can replace "to be" in a given sentence with either "is part of/equal to" or with "is something that exists". That way I can see whether i have to use copula (desu) or the verb for existence (imasu/arimasu). Also, I would generally recommend thinking of "imasu" and "arimasu" not as both "to be/to exist" but only as "to exist". That way you don't confuse it with the copula meaning of "to be".
Btw, the same goes for your second example sentence "Isu ga arimasu", which translates to "A chair exists" or "There is a chair". Lastly, you wrote "arimaAsu" - you can leave the extra "a" out:)
To say "It is a cat", you would use です, not います.
So it would be ねこです "It is a cat" and いすです "It is a chair". In contrast, ねこ が います。 means "There is a cat" and いす が ありまあす。"There is a chair"
These are different sentences and which one you want depends on what you are trying to say. If someone asks "What is that?" you might say "That is a cat". If someone asks, "What is in that box?" you might say "There is a cat." Be careful to not confuse these two sentence patterns.
Also, watch out for the polite negative form ofです. It looks a lot like the polite negative form of ある。
ねこがありません "There is no cat"
ねこではありません "(It) is not a cat"
Look for the では. It makes a huge difference.
Damir1899, You are right about it sounding wrong. The answer should be "I am busy until THE WEEKEND (only the singular is correct). You should report it. In English, to use the plural, I would say, "I am ALWAYS busy EXCEPT ON weekends (This one can be with or without "the", whereas the first one related to this discussion HAS TO HAVE "the"). Here is the Japanese for the second one: Shuumatsu igai wa itsumo isogashii desu. しゅうまついがいはいつもいそがしいです Of course this is a different meaning than the sentance of this discussion. By the way, good effort in English!!