Temp, 225 or higher. If Thick 7-8 ounce burgers, you can Smoke them to an internal temp (IT) of 120°F, then sear them off on the Grill or hot Cast Iron Pan to 135-140, medium. For thinner burgers, you can Smoke an hour then sear but well done will be your only a option.
Time, depend on thickness, 1 hour or so, you want to be able to sear without drying them out.
Wood? Hickory or Oak is good. If you have experienced Mesquite and like it, you can get that Texas punch on them Burgers...JJ