Yes, it's correct. It's not uncommon to have shortcuts in Japanese, especially if the connecting sounds are phonetically similar-ish. Casual masculine Japanese is particularly guilty of shortening up words and phrases, but it's common for both sexes.

Yes. The term you're looking for is い抜き. It's one of those things that's technically, grammatically wrong and improper, but has been getting more and more common in casual speech, along with dropping "ra" or needlessly adding "re".

Example a random business blog: https://i-career.co.jp/blog/manner/inuki-ranuki-reire/

ゆっくりしていってね！Japanese uses both して and していって (can sound like してって). Adding いって to something gives it a sense of being "ongoing" and other forms of て+いる also exist