In short, go with flat runes if you will have a significantly stronger laning presence because of them. Although per-level runes outpace flat runes around level 13 (sometimes even earlier), the strong early game presence provided by flat runes can out-value the late game presence provided by per-levels.
To answer your questions more directly:
Certain champions that have high stat scaling or activated abilities use flat runes to significantly empower them early game. Some examples:
EDIT: This has since been changed but I'm saying it for historical reasons: Akali's Twin Disciples passive used to provide 10% spell vamp and 10% bonus magic damage when she has a bonus of 10 attack damage and 20 ability power, respectively. Having this at level 1 or 2 is very appealing, and this is what pro Akali players tend to do, including the top Akali Westrice.
Champions with near 1:1 ability power to damage ratios on their abilities (Sion, Amumu, etc.) can turn an early game 50 ability power directly into 50 damage, which is scary for early levels when you have from 400-600 health as a squishy champions. You will have to be weary of taking damage from them as this will badly hurt your early game.
Crit builds, mostly crit Gangplank, rely on flat critical strike chance or damage early game to win the lane through big critical strikes when harassing. Anyone who's had the unfortunate experience of being killed by a huge crit during a low-level duel knows how strong this can be, albeit very luck based.
You should never buy less than Tier 3 runes. Tier 1 and 2 runes provide small bonuses and may be responsible for at most 3-5 of your low level wins, but once you hit level 20 you'll be glad you saved your hard earned IP for Tier 3 runes. Runes cannot be bought with RP so you'll be forced to grind for them, which I've been doing for the last 200 games and I'm still not finished. You will still win plenty of games without a rune page before level 20, speaking from personal experience.
There is always a certain level at which per-level runes outpace flat runes. That level is usually around 13, but as stated in this answer some outpace at level 6. Almost all of the time you need to measure the utility you get out of your runes by yourself, because their value is largely based on your skill and style. The exceptions I know are that flat Armor, flat Magic Resist, flat Attack Damage, and flat Ability Power Quintessences are are far more valuable than per-levels because of how important those stats are early game. Early game farming and presence is very important, which you will learn with time and I won't cover here. Flat Mana Regen runes can also help early game sustain.
Referring to playstyle:
In terms of offense, flat runes are really worth it if you can use them to force enemy champions out of their lane and zone them early game. For example, if those flat runes get you three waves of creeps and 1 more kill than your opponent, you have a ~800 gold advantage that more than makes up for the stats lost by taking flat runes over per-levels. This is most commonly done by killing your lane opponent and pushing the lane to their tower.
In terms of defense, I always use flat runes to counteract the power of others. If your opponent happens to use a strong combo of flat runes on a champion in most cases you will be crushed in lane (read my answer to #1). By starting with flat runes like Armor yellows and Magic Resist blues I get a great +13 Armor and +13 Magic Resist. Health Quintessences are also decent. They mean you can basically ignore enemy champion harassment for the first 5 levels and farm as you please. There is nothing more discouraging to your enemies when they can't force you out of lane after trying their hardest.
I've racked up a lot of Dominion games and came out deciding that my advice on flat runes still applies to Dominion, if not more than in Classic gameplay.
This sounds odd given champion level increases so quickly in Dominion, meaning per-level runes reach their potential faster. However, per-levels are only viable if you can farm properly and prevent your opponent from killing you. In Dominion, there's no consistent farming and you have to fight! If you're staying in lane farming you're likely going to fall behind everyone else who is focusing on getting kills and capturing points. So you can't reasonably expect to avoid fights and just farm. You're going to be fighting opponents with all-in flat rune strategies, and with per-levels you're likely going to be losing.
In particular the initial fight for the Windmill happens at level 3. At this level flat runes still provide more power than per-levels, so they are important to have if you want to win that fight. If you don't win that fight you tend to lose the whole game - all 4 of your teammates dying at the Windmill is a ~500-1400 gold advantage that the enemy team has (depending on how many of them died), and with that gold difference you can have a very hard time catching up.
Seems like Riot has targeted flat runes as being a source of balance issues. Flat magic resist has been nerfed because of the huge advantage you get from stacking MR as mid lane, so the per-levels are actually very appealing now. Be on the lookout for more flat rune nerfs because if they are significant enough (say, per-levels outvalue them at level 4 instead of 6) then my advice doesn't apply to that rune.