Great clues as always. I particularly like number 2- fiendish. I’m not 100% sure of my answer to 3.
My suggestions are:
1. Spring chicken register (5,2) CHECK IN. Anagram of chicken (spring is indicator)
2. Cut down and captured, grabbing criminal closer (5-7)CLOSE CROPPED. Cut down is def. Anagram of closer (indicator is criminal) inside copped (caught)
3. Going from coin-teller to successful speculator (5-7) MONEY SPINNER. Coin=money and teller as in story teller=Spinner
4. Bishop abandons a workhorse to grass (3,2) DOB IN. Dobbin (workhorse) missing B for bishop. Grass as in inform on.
5. Old British man with socialist wit is attached to pioneering transport (7,5)COVERED WAGON. Cove (old British man) Red (Socialist) Wag (wit) On (attached to).