"1. Can the てお -> と shortcut be used for contexts other than ておく (for example: お世話になっております -> おせわになっとります)?"
The なっております-to-なっとります contraction does happen dialectally. You will hear it many times daily in Central Japan and Kansai. I am sure that it is used in many other parts of Western Japan as well.
Around Tokyo, you will rarely hear it used. When you hear it, the speaker would usually be someone from Western Japan or Central Japan. The ておく-to-とく contraction for the meaning of "doing something in advance", however, is used heavily around Tokyo as well.
"2. Are there other 2-kana to 1-kana shortcuts? (ていう to って, い抜き言葉 and ら抜き言葉 are interesting shortcuts, but the idea of shortening 2 kana to 1 kana sounded interesting to me for some reason.)"
Yes, there certainly are. I will include the ones with reduced syllable counts if not reduced kana counts.
ている ⇒ てる ([食]{た}べている ⇒ 食べてる)
でいる ⇒ でる ([飲]{の}んでいる ⇒ 飲んでる)
ていく ⇒ てく ([見]{み}ていく ⇒ 見てく)
でいく ⇒ でく ([休]{やす}んでいく ⇒ 休んでく)
ては ⇒ ちゃ (なくては ⇒ なくちゃ)
では ⇒ じゃ ([泳]{およ}いではいけない ⇒ 泳いじゃいけない)
てあげる ⇒ たげる (キスしてあげる ⇒ キスしたげる)
I feel like there should be more.