#1 - try out a 57, 58, 59 and 60s reissue if you can. I don't have big hands either and none of these models have baseball bat necks. They're all comfortabe to me. About the only thing I don't like are really thin necks like Ibanez Wizard type necks.
#2 - Custom shop models are more expensive, but totally worth it if you can afford it. Better finishes, better craftmanship, etc. Don't overlook used custom shop LPs. You can find great used reissues for $4K or less.
#3 - Burstbuckers are Humbuckers. I personally don't like Burstbuckers in LPs (but they sound great in my 335), but a lot of people like them. I never worry about what kind of PUs come on a LP because they are easy to change out and really, Burstbuckers are perfectly fine. You're amp choice is way more important. I look at playability (action, comfort, how well it sustains notes) and the quality of the workmanship. You don't even have to plug it in to test that.