Does weight and climbing cause strain on bike?
The spokes and gears are likely unrelated issues. 
You have done 2000km on the bike - have you replaced the chain and rear cluster yet? At you weight and those hill climbs, I would not be surprised if they are just worn out. It is also possible that at your weight and that distance the spokes have come out of adjustment, and/or now have fatiguied to the point of breaking or stripping threads on the nipples. I would also be checking for a broken axle and the bearings not being adjusted properly.  

The bike shops should have picked up all these issues, however, are they aware of the distance you are riding? Many such bikes do 50 miles a year and they may be servicing it based on that assumption. It is a mistake to presume that any LBS is competant, servicing is and art and a passion, and (putting it politely) some mechanics are better than others.  

What to do - ask around for recommendations on bike shop. If you are comfortable, take it back to one you have used, but make sure you let them know how much use the bike has had. Talk to the mechanic directly, don't be fobbed off by front counter salesmen (And walk out if you cannot get access to the mechanic). Ask him to measure the chain and check the cluster and chain rings. Check the spokes/bearings and axle. Be prepared to replace parts.