The battlefields are areas with randomly generated dungeons, as well as several merchants. The dungeons are entirely optional (I skipped them), but having handy access to merchants whenever you wish is nice. Merchants can hang on to some stuff you sell them, which stay at the top of their list, so I tried to sell most of my stuff to the potion dealer, since I rarely wanted to check out her inventory, anyway. Other than that, their inventories reset each time you enter, so if you want to look for equipment upgrades, you can quicksave, enter the BF, check merchants and quickload if they done have anything. I wouldn't waste too much time on that, unless you really start having trouble.
Have you found large keys, that when clicked on, say you access to battlefields level x (1-3 in the first act)? There was one in a mostly hidden locked chest on the first level, with a key hidden near the end of that level. The second (level 1) key you would have passed after the second maze level, before getting back to the first level and access to the citadel. The level 2 key can be found in the citadel, in a side room in the area with the DK patrols. If you get the level 2 key, it will open dungeons 1 and 2, so there is no need to backtrack to get the level 1 keys. You can even wait until after you confront the demon Samuel near the end of the act, after which you can find the level 3 key.
If you have clicked on a BF key in the current act, when you click on the secondary skill button beside either characters' portrait, there should be an 'Enter Battlefields' icon at the bottom of the list. The first time you enter in an act the dungeons are generated, so it can take awhile to load.
Speed helps determines your evasion and initiative. Survival helps determine initiative, luck, regeneration, accuracy and elemental resistances (see the strategy guide, math appendix for details).
Speed effects your reaction time in combat (the Reload Time bow subskill is a straight Speed boost, for example). Heavier armour will make you slightly slower (not sure it would be noticeable visually), but the better protection more than makes up for that. You can also find jewelry, and sometimes armour, with +X% speed bonuses.
The prisoner wants a specific bottle of water (it updates the quest log when you pick it up) from the kitchen south of there.