What are the proximity bonuses for adjacent buildings and when do they apply

You can find a detailed overview of the differences between all four difficulties in this thread on the 2K forums.

Here is what that thread says about Normal and Classic:

Normal:

1. Base global panic starts at 0
2. Most minor alien activity raises panic by 1 in left unchecked with some events doing up to 3
3. Your soldiers have no bonus hitpoints
4. Aliens have their AI mostly unshackled
5. Soldiers cost $10

Classic:

1. Base global panic starts at 8 (1 per location)
2. If unchecked alien activity causes at least 1 panic, but will often cause 2 or 3
3. Your soldiers have -1 hitpoint
4. All aliens receive bonuses hit hitpoints, accuracy, critical change and / or damage.
5. Alien AI is unshackled
6. You don't receive officer training school for free. You must unlock and purchase it.
7. There are even more aliens per map
8. Soldiers cost $15

The big disadvantage of sprinting only happens when you do so into unknown space. Sprinting into an area, only to uncover one or more "nests" of aliens is a quick way to get your face plastered onto the memorial wall.

In most cases, you want to split your move up, just in case your first 'half' of a move changes the tactical scene. There is one benefit to sprinting, though -- reaction fire takes a -20 penalty to Aim against a moving target. It's not a benefit you'll see often, but dashing into safety might be the best option if one of your squaddies is surrounded by hostiles, or being suppressed.