1) Is it allowed in Islam to lit the AGARBATTI before a grave and also inside our house while reciting darood-e-shareef.

2) can we do Fateha in the name of Muhammad sallal la hu alaihi wasallam in the month he was born.

Thank you.

No lighting agarbatti in front of graves is not allowed. In fact any kind of grave worshipping is not allowed in islam. It would be consider as human worshipping. Hence praying on graves is non islamic. One also not light agarabatti in home for prayers. Instead use of rasin is allowed.