Hair Oiling is not shown to help hair growth or prevent hair loss. I would recommend waiting another 1-2 months to see if you have graft growth as shedding of the grafts is possible during the timeframe you have mentioned.
If you are having hair loss, you need to see a diagnosis. For men, the most common (99%) have genetic causes of the hair loss which is easily treated either medically or surgically. A doctor will tell you what is best,
Hair loss is progressive and you may be losing your native hair. to slow down future hair loss you can start on propecia or minoxidil.
Oily hair or oily scalp does not harm transplanted hair. It usually takes 6 to 12 months to see the results after surgery.
Don't panic! Due to the natural cycles of hair growth, transplanted hair typically sheds 2-6 weeks post operatively. This is normal. The hair then goes into a "resting phase" called telogen and will start to regrow hair 4-6 months post-operatively. Your hair sounds like it is going through this normal cycle and should start to regrow visibly between 6-12 months.
Hair oils are often used in different cultures but there is little evidence that they help hair other than to make it look shinier. It is NOT likely a factor either in your graft growth, or your male pattern loss in the first place.
Hope that helps!