"There are seven tables."
「あります」is a verb, the polite form of the verb 「ある」which is an existence verb for inanimate objects, often translatable as the English expression "there is".
「ます」 is the ending that is used in the polite form of almost all verbs, an exception being the verb "to be", which has the polite form 「です」.
I personally found this easier to understand when learning casual speech first and then moving to polite forms, which is the way most native Japanese speakers learn (people usually speak to kids in all casual speech and young kids usually speak back in this way, from what I have been told by native speakers).
For some reason most modern courses teach the polite forms first, which seems a little unnatural to me. It's like, yes, it's good to be polite by default, but it's not the simpler linguistic form and not as clear for teaching the language.
卓（たく）is supposed to be the table and desk specific counter. Not currently accepted on Duolingo.
Tofugu claims that 台（だい）(counts large machines and things you put things on) and 脚（きゃく）(Counter for legs and things with long legs) should both also work, but again they're not currently accepted here.
Nothing grammatically i dont think. I do believe this is an answer to a question though. 七つテーブルはあります would be a way to inform people there are 7 tables. "7 tables" is the important information as a combination word. While テーブルが七つあります is the answer to テーブルはいくつですか？You can probably use both you'll just sound a little off.
In short 7 tables first is "as for seven tables, there are"
7 before the verb "as for tables, there are 7"
According to the counters article on iknow.jp/courses/745749, the -個 counter is used for small, circular objects like apples. The -つ counter is more of a generic catchall, they say it's better to use the proper counter if you know it, but some categories are archaic and no longer used - see SIK3E's comment. I've heard native speakers use the counter -つ for tables and chairs.