>>1013123The position of the hoods on your bars can be adjusted by first loosening the bolt that's recessed inside the lever body (accessible either behind the lever blade, or it's built into the lever body and is hidden under the rubber hood). Some, but not all, levers also have a reach adjustment screw (built into the lever body, points towards the rider) that can be used to change the default distance between the lever blade and the bars. Show/tell us what levers you have if you want more details.
>>1013156Yes, you need a shim, they're pretty common. If you install the post without a shim you'll damage the seattube.
>>1013163BMX hub in an old road frame? Make sure the axle is going to fit in the dropouts before you go any further with this project.
Most LBS won't be willing to 'cold set' a frame, they'll probably tell you to take it to a local frame builder if there are any. It's not very hard to do your own cold setting, but with that much of a change you'll also need to DIY dropout alignment gauges.
>>1013138TRP Hylex. Also, learn to google you lazy fuck.