やる is the dictionary form. It is an う-verb, not a る-verb. You can tell it's not a る verb because they normally have いる/える endings (たべる。 みる。 there are some exceptions which are still う-verbs, such as 切る(きる).

When putting an う-verb into ます form, you bump the う into an い。

So やる becomes やります まつ becomes まちます のむ becomes のみます

Etc. Etc.

When you're turning a る verb into ます form, you just drop the る。 たべる becomes たべます みる becomes みます ねる becomes ねます

Does that make any sense at all? I'm not much of a teacher haha.

やる is the base form of the word. やり is the polite form when paired with ます. So やる is just "to do" While やります is "to do (polite form)"

From what I understand, やる is the dictionary form and やります is the polite form. Because it's a U-verb, –る becomes –り and +ます is added to make it polite: やる → やります

Yaru is a verb, similar to suru, and if you want to use yaru politely, it turns into "yari"masu. So yari is just the masu-stem of the verb yaru, just like noru becomes "nori"masu, or kiku becomes "kiki"masu.

Either way works

Both ways work, but saying the time first sounds more natural, according to my teacher.

まつ is end. If you just say 週に, it will mean during the week instead of during the weekend.

引きます hikimasu 弾きます hikimasu

It actually means to "pull" or "draw" or "flick"(stringed instruments), but since nobody talks like that in English, it gets translated to "play".