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
ようこそJapaneseLanguageへ！ 日本語に興味を持てば、どうぞ登録して勉強しましょう！日本語に関係するどのテーマ、質問でも大歓迎します。 This is a community dedicated to the Japanese language. Feel free to come in and ask questions or post your thoughts and opinions about this beautiful language.
Feel free to check out the web archive of r/LearnJapanese's resources if you're looking for more learning material or tools to aid you in your Japanese language journey!
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Remember that you can add furigana to your posts by writing ~{KANJI|FURIGANA}~ like:
~{漢字|かんじ}~ which comes out as:
{漢字|かんじ}