Hi there. Without any further information it is difficult for me to provide the best answer, though I will certainly still try to help.
His symptoms of lethargy, reduced appetite and pooing outside of the tray would not be specific to any one issue and could have multiple causes including:
- A food allergy or sensitivity
- Inflammatory Bowel Disease
If he has not had a recent vet check, that would be the first step. We may also wish to have his stool analysed.
If he is the only cat in the home he should have 2 trays which are in quiet areas of the home and cleaned regularly. Cats often won't use a tray that has waste in it. Try different litters, as some cats are fussy about the ones used.
Monitor his weight to ensure he is not losing any.
Ensure he is up to date with a good quality wormer such as Panacur.
You may also wish to try a hypoallergenic or sensitivity diet.
To reduce the risk of fights with the other cat, consider keeping him indoors when that cat is out. If it is a stray/feral, you should be able to contact a charity such as the RSPCA who could help him. They often neuter these 'problem' cats and he may even be re-homeable. If he already has a home, consider discussing the issue with his owner and suggesting he is kept in at certain times so your cat (and likely others on the street!) have a chance to get some fresh air and exercise without being bullied.
I hope this helps!