1. When is it time to push? How do you decide which lanes to push?
Depending on team comp, right away, always after a enemy Champion Kill where you have a number advantage in lane. If you can't get to a lane to help against a gank, but can counter-push. Always get tower hits in when you can, even if that means taking a few potshots in the process.
2. When you're team battling, who should be targeted first? Should you hit the stuns? The DPS? The tanks?
This is the most difficult thing to decide on, as typically the one you want to target first is a Shaco or a Twitch that wont show up until you engage with the rest of their annoying team, You should always first burst down the squishy dps target that is the closest, or beat on tanks that push. This is very frustrating to do and coordinate as positioning and where the battle takes place is always a factor. A team with a little map awareness, and on your end pushing towers seems to always be more difficult to engage and focus on then when you are on their end, as by then there is typically an advantage for them. If it is still anyone's ball-game try to single out their ranged dps, then melee dps, depending on the threat, tanks can be mostly ignored, but if they over-extend punish them, and their fragile team is left behind to pillage. (I have a bad habit as a tank of overextending)
3. When should your team stick together as one? When is it safe to split up and work on multiple lanes?
Usually at higher lvl play, it is always a mistake to split push lanes unless you have vision of your enemies, all towers are still up, or they persist to split push. This is usually after the side lanes hit lvl 6-7. Some characters can split push very effectively. Like a Sivir/Shaco/Twitch/Yi provided their movespeed and ability to quickly take down towers with or without minions.
Ive also seen Trynd, Nunu, Kassadin and Karthus do this well. Primarily you want to start 5man map control around lvls 6-7, taking frontline towers asap before they take yours, and taking over the jungle to bleed out their experience gaining ability.
4. When should the team work on the dragon or the baron?
This is such a crucial aspect of winning games, that it drives me nuts during a game. I usually am the one with Clairvoyance, and eye this spot so many times if my team refuses to go and grab it asap I usually know it is a sign that we will be in for a long painful game. You need to fight over dragon, and take Dragon at every opportunity. A Shaco with a Smite can turn-games around be stealing this kill. The Dragon gives global gold and EXP, the EXP is what makes it game changing. The last game I played last night, we had completely lane dominated them, but they took dragon twice by lvl 8 and out leveled us. That slight level boost allowed them to out push us 5man and ultimately win the game, after we had out killed them 8-1 in the first 8minutes of the game. Another example is the game before that, where I pinged Baron, knowing they were going for it, and 3 of us were there, 2 running from base. I stepped a bit too far out, and they took me out quickly, not getting the support from my nearby teammates, and then they ran back to base afraid to engage. Once they had Baron, it was very easy for them to push our inner towers, as the buff offered enough to push their team over the edge, and the global xp+gold helped them hit max faster. Watch out for Baron pushes around lvl 12 now, as it can be and has been done many times that early. Ward these places as often and as much as you can.. Mushrooms, wards, Jitb's etc.