1) Depending on your level, Mage's Ballad restores about 40-80MP per tick (Battle Voice improves it greatly). It is useful in situations where your healer(s) are spending lots of MP and need the song to keep them from running out. SMN may benefit from this, too, since they'll have more MP to cast their attack spells. Don't worry about using this song for BLMs, though. They have a great tool for MP restoration.

2) Yes, but not just Army's Paeon. Both Army's Paeon AND Mage's Ballad affect you. Mage's Ballad lasts the longest of the three songs since it restores your MP, and Army's Paeon (plus Battle Voice, if you have it) is great for TP restoration for the entire party. Foe Requiem only affects monsters, reducing their magic resistance by 10%. Be careful when it is up, 'cuz you can accidentally aggro monsters with it. Make sure to turn it off after each pull.

Here's a simple rundown of when to use your songs, along with the Battle Voice CD.

Mage's Ballad = When healer(s) runs low on MP.
Foe Reqiuem = Whenever you are paired with caster DPS (BLM/SMN). Feel free to use it liberally if you have caster DPS in the party and MP to spare as it doesn't decrease your damage when it is up, but turn it off after each pull because it pulls hate from un-aggro'd enemies that are within range. Wait until the tank gets a good hate base before playing it, and turn it off when all enemies in the pull are down.
Army's Paeon = When party is using (or spamming) AoEs that cost TP.

Battle Voice with Ballad = When healer(s) run dangerously low on MP, need to revive multiple fallen party members and heal those alive, or spamming AoE heals (i.e. Binding Coil of Bahamut Turn 2 Enrage Strategy).
Battle Voice with Requiem = To really push caster DPS/turn those Flares into true mini-nukes.
Battle Voice with Paeon = To keep up the AoE spam (i.e. Binding Coil of Bahamut Turn 4).

And there you have it. I hope this information is helpful to you, and good luck playing as a BRD!