Thanks for these great clues, Jolan.
A delight as always.
Here is my stab at them.
1. Goes around rotisserie to find long-bodied short-legged dogs circling the meat (9) TURNSPITS?
Goes- turns around rotisserie – spit gives turnspits - long-bodied short-legged dogs used to keep the roasting meat turning?
2. Old Norse drum kept in cask (2,3)- ON TAP?
On (Old Norse) + Tap – drum gives “on tap” – kept in cask?
2. Time being the only non-cellular occupant (5) NONCE?
In clue. Nonce meaning "time being" as in "for the nonce"
3. Carefully done notice, outside job centre, leads to first employment (6)- SOIGNE?
Notice- sign + o (centre of job) and first of employment – e gives soigne – carefully done?
4. Toss back drink before playing with toys of a prohibitionist (9) PUSSYFOOT?
Sup- drink , tossed back , Pus, with anagram of “of toys” gives pussyfoot- in Chambers as a prohibitionist?
Thanks again,
AB