The earliest I have found that you can unlock a helicopter is during the main storyline. Once you complete the mission Trojan Whores, you receive a Vulture helicopter for the mission reward. It's accessible at the Saints HQ pad in Sunset Park. Just go outside next to the pool (after you're inside the HQ, so the outside of the penthouse) and you'll see the floating key indicating a garage, which is where you'll be able to store future helicopters.
I was level 15 when I acquired this, which is 9 levels earlier than you can receive the Heli Homie. I'd also done a bunch of activities on the side to raise my level though, so if you just go straight for the Trojan Whores mission, you'll unlock it earlier.
I would recommend doing everything in your power to finish the level as quickly as you can. Don't worry too much about trying to earn a lot of money here, there are other things you could do in the game to get that.
To be able to do that, that means getting the money to be able to leave as fast as you can then race to the finish. You'll need to have the right weapon upgrades and you need to be a crack shooter (mouse/keyboard combo is a necessity here if you have the PC version).
Accuracy is very important here as you can build up the multiplier for every kill you make but lose it if you ever miss. I would recommend you come in equipped with any pistol at level 4 with the (optional) akimbo pistols upgrade and use this weapon exclusively for the first half of the level. Akimbo pistols effectively doubles your ammo between reloads so not essential here.
Go for headshots on all the mascots. One shot, one kill. Try to do it as quickly as you can and don't miss! Don't worry too much if you get hit, take your time, you can almost always find a health target nearby, just don't take too much hits. When you have a high enough multiplier, go for the money shots for maximum effect. When you have enough money, go ahead and switch to your strongest weapon (if you are able to earn presents) and go on to the finish.
When you encounter the brutes, it would be better to avoid them and not kill them (especially the first one). As you know, they will respawn endlessly if you do. But don't just try to run past them, you'll need to damage them enough to the point that they cannot move anymore (and you have the choice to finish them off). Once they're in that state, then you can move on past them.
Otherwise, kill anything else in your path to the finish.
Best Answer
You can store cars in the garage at any crib, and at any Rim Jobs. Cribs show up as little house icons on a yellow background. Rim Jobs are green icons with wrenches on them.
At the start of the game, you'll only have access to Shaundi's Ex's Loft as a crib, but complete more story missions and more will open up to you (as Ashley list indicates). Some of the cribs have helipads, and these store air vehicles such as choppers, and later... well, I don't want to spoil it ;)
You can buy Rim Jobs locations, but this is not required to store cars or access your garage. This just allows you to lose your wanted level if you enter it, and gives you a "periodic income" boost. You'll be taken to a Rim Jobs location early in one of the plot missions, but they are located across the city and are unlocked from the start. Rim Jobs locations can't be used if you have a wanted level, unlike crib garages which work regardless of your wanted level.
If you invest any money in a car, it becomes yours permanently unless you delete it from your garage - you can lose the car, destroy it, drive it off a cliff into the river, etc, and it will always reappear in your garage shortly afterwards. Under certain circumstances you may have to pay a $500 fee to recover a car that's been lost or destroyed.
You can usually store cars whenever you wish, so long as the car isn't required for a mission. There are times when you can't access the garage, and the game will tell you so if you try to approach it.
When it comes to what can be stored - if it is a land vehicle, and you can drive it, you can store it in any garage. You might not think you can fit a tank inside Shaund's Ex's Loft, and it's a little bit of a squeeze, but it fits :)
In addition to the "land vehicle" garages, and helipads at certain cribs, there is also an airplane hangar at the airport you can buy, and a boat dock or two if you wish to store some boats.