Screw extractors, DON'T!

Next time weld a nut to the broken stud. Jody has a pretty good video on that.

Another option is to cut a chunk of aluminum to fill in the missing material, that will reduce the amount of weld and heat you will put into the part.
Either way from how it's broken I would probably leave it together, at least while you weld it. After you're done make sure that valve and rocker are free. I probably wouldn't do any preheating, the part will get hot enough from the welding. As Zoama said, give it some cooling off between passes. Once you can put your hand on it, do another pass. Make sure to clean out all the junk in whats' left of the threaded hole. I would open that part up with a rotary burr, so you are not trying to put new threads right where the weld meets the parent metal. Your biggest issue might be the quality or lack thereof in that casting. From your cut edges it looks like it's pretty good material. I've welded quite a few of those Honda knockoffs, and while some are great, others are really poor castings.