When doctors try to manually push back non-bleeding haemorrhoids back into the anus, do they typically use their lubricated latex gloved finger directly? will this process resulted in a torn & bleeding haemorrhoid?

Yes they do and if performed properly will not result in any damage.

Yes they SHOULD use a lubricated finger gloved. The real concern is if this is a true external hemorrhoid it should not be pushed inside as it doesn’t belong there. If it is a prolapsed internal hemorrhoid, reduction internally is a good first step followed by advice on avoiding straining, constipation, diarrhea and sitting on toilet for more than 5 minutes to have a BM. See a Colorectal surgeon.