I recently had a very similar shopping list as we've just bought a villa in PB. Got some very nice metal framed sun loungers from Home Zentrum in PB. On the small ind. est as you leave PB take the old Arrecife road.
Got the TV from Electron which is a few doors up from Ikea in Arrecife. Great shop - loads of choice and well laid out. Only thing to be aware of is that you may need some ID if paying with a card....
BBQ - i ended up getting just a cheap one for now from FT's on the trading est. but I really want to put in a decent brick one at some point but not sure of the best place to pick one of those up. Let me know if you find somewhere