your lazy engagement from cold is down to the clutch seals they have gone hard and from cold the fluid is blowing past them, when it gets a teeny bit of heat in it the seals get a bit more plyable and the clutch engages you get drive, the missed 2nd gear will be a similar problem either the front servo seals are bad and the piston just wont engage or the band is goosed could be adjustmnent but doubt it, front band adjuster crack of the lock nut screw the center stud in till it nips up back it out 2 turns lock the lock nut its not to fussy if still no 2nd its history tranny out time,,,,,,,, hope this helps
Watts racing transmissions england,,,,,