Since you're savvy enough to check out the diaphragm, I would suggest replacing it. I recently bought one for a B&S 550 engine I came across. If you do any carb work, it's cheap insurance to replace the diaphragm, the one I got was $3.90 shipped and it's an OEM part, not aftermarket. Since yours has tears, I would start there. If that doesn't solve anything, you spent $3.90 to totally eliminate the diaphragm from the equation. Before you install the new one, take a can of carb cleaner and squirt through ALL the holes in the carb until you clearly see the liquid come out another hole. If this is the same engine I worked on, there's no adjustment on the carb so there's more guesswork eliminated right there.
Aside from that, does the engine run on starter fluid? I wouldn't worry too much about the rope. I had a 6HP engine that only had half it's pull rope and I used it like that until I could get new rope. Didn't have a problem because the engine ran fine. It's just difficult on the shoulder

Once you get the engine running, you can fix up the rope.
*EDIT*
For reference, here's the diaphragm I bought for the work I did:
http://www.ebay.com/itm/25085935168...NX:IT&_trksid=p3984.m1439.l2649#ht_500wt_1180
Great seller, fast shipping and, most importantly, OEM. You should double check that the part # matches the one required for your engine.