Re: Mountain or Cruise?
Personally I recommend a cruiser for around town riding. My first was a mountain bike and it got sold quick. I just bought the motor for the fourth bike I am about to build.
At the $200 range, I like some of the multispeed Schwinns available from walmart. Specifically the Clairmont and the Landmark. There are others in that range, depends on what color you like and if you want a rack. Shpould be a topic here not too long ago marking the differences. Mostly color schemes and if it comes with a rack or not.
I just ordered a Marina Fito from bikebuyers, with a $20 light for $234 shipped. Looks like a really decent bike with nice parts. Funny enough, doesn't seem anyone has ever put a motor on one of these. Couldn't find any mention of them on the site.
There are 2 schools of thought when regarding a coaster brake on cruiser, Those that like them and those that don't. You may not use the multispeed on a beach cruiser. But every one I have seen comes with front and rear brakes and no coaster to modify. Saves hassle and buying a front brake which most people will recommend.
As far as the engine fitting, I have never used anything but a 2 stoke. You'll have to check elsewhere if it will fit or not.
Good luck, and welcome to the obsession.