While its possible that you do have some pattern failures, one thing is certain your O2 sensor is not the cause.
First why its not.... The O2 sensor doesn't do anything untill its hot and even if the input was created wrong it would have no part in starting enrichment.
The pattern failures that apply to your symptoms are: warm-up pressure regulator clogged or inactive leaving high control pressure. This accounts for the cold running that disappears when warm. The hot starting problem has another pattern failure: fuel pressure leak down with subsequent fuel boiling/vapor locking. Much on both in the archives.
Both of these problems are simply identified with proper fuel pressure testing. The gauges can be expensive so you might want to find a technician with the equipment and more importantly the knowledge.
Bosch Master, ASE Master, L1
33 years MB technician