What size timber should I use?

I would use 4x2 treated cls for plates and joists as long as there is no weight on the new ceilings above. Either fix the wall plates with 4 inch screws or bolts into the brick ,if the brick is not very good the use chem fix with a threaded bar and bolt the plates on, for the lathe and plastered wall there should still be a solid wall behind if not then find the studs and screw into them. Hope this helps.

Probably 3"X2" CLS timber you should be able to punch holes through the old ceiling to find joists and screw timbers to them to support the new ceiling joists .fixing to the lath and plaster wall shouldn't be a problem just keep putting a screw in along were the new wall plate is going untill you find a upright

I agree with SK carpentry once the joists are on I’d put noggins in the centres to make the frame ridged and if required some vertical timber hangers tiring the new ceiling into the old.