A bit out of practice: flying from Stockholm
to Heathrow on BA then Heathrow to Dallas on AA. All one ticket. About 2.5 hours to connect.

once I deplane, I follow the purple signs and take a bus to terminal 3, from 5? Do I go through security again? Customs or immigration?

Related, how early do I need to arrive in Stockholm airport before flight to London? Carrying on only.

You will go through security. (everyone transiting through LHR whether same or separate terminals has to go through security). No customs or Immigration as long as you stay airside.

And I should take busses and follow purple track? Or tube?

If you follow the flight transfer signs and take the transit bus you are staying airside and don't go through immigration.

If you go through Arrivals to take the Tube you are actually entering the UK so you will have gone through Immigration.

Either way is allowed but 'normally' staying airside / using the bus is faster

Definitely follow the Connections signs. Is your AA flight on their plane or double badged and actually with British Airways?