When you're last hitting in Dota2, a lot of the damage from creeps is being dealt by the ranged creep. Notice which enemy creep it's attacking and often simply just time your projectile to hit immediately before or immediately after the ranged creep's attack to ensure the last hit. You should be hitting each enemy creep exactly once- this way your enemies won't have a chance to deny it while you're waiting for your next attack. If your opponents are too close to the wave, especially if they're melee and you're ranged, you can harass your opponent to completely remove the ability for them to deny you. Doing this will draw creep aggression to you, however, so you should usually start to back away as soon as your attack is released to minimize the damage you'll take (none, if you've positioned properly). If you feel like you're in a good position, you can also animation cancel (begin walking towards your target) as soon as the attack is released, and usually get off more attacks and really harass your opponent. Just be careful when chasing past the creep wave, you usually don't want to expose yourself that much unless you're in a strong position and an creep wave can do a surprising amount of damage.
Your starting items will also help you to last hit in lane dramatically. My typical ranged carry build is a +3 primary stat, a +2 all stats, and three +1 all stats items, and a health potion or tangos. This gives you adequate health regeneration, +95 HP, and +8 damage, and builds nicely into a Magic Wand and a Wraith Band/Talisman. For melee carries, I usually buy a Stout Shield, a health potion, tangos, and three +1 all stats items. I see a lot of new players buy boots or other worthless items first that don't help them win the lane, they just lose it less dramatically. You'll be surprised how simple choices like this can help you crush a lane early against equally skilled opponents who are making bad choices. If you're a melee hero competing with a hard hitting melee hero or an opponent who keeps denying creeps, consider buying a Quelling Blade to improve your damage. If it lets you get 3 last hits you otherwise wouldn't and then you sell it, it's more than paid for itself.
One concept calling animation canceling is unintuitive at first but incredibly important both in lane and in general. To perfect your last hitting, you'll need to know how to do this and understand your hero's attack point. Here's a full list of Attack Points that's sortable (add the Atk Point column)- they typically are between .3 and .6 seconds, with a few outliers. A lower attack point is better because it lets you prepare an attack faster. It's kind of like riding the edge of a trigger: you want to be really close to firing as often as possible, but not actually fire until you're sure of it. When you know your hero's attack point (which will improve over the course of the game as you gain attack speed), you can start attacks and cancel the animation just before it completes by pressing S or H, immediately being able to start another attack. Doing this repeatedly is very effective for faking attacks and causing your enemy to misplay- they won't know when your true attack will be coming out. To last hit with ranged heroes you'll also have to account for projectile speed, which also varies but can be seen in the same link.
Everyone usually shares a single courier that a non-carry player should buy at the beginning and upgrade to flying as soon as possible. You can buy more couriers (as many as you want), but most teams only use one. In general, solo mid gets first rights to courier use, especially in order to get their bottle.
Other than that, it is generally not encouraged for you to bring items to yourself that you could buy at the side or secret shops, as they are close enough you can just walk there from lane. If you do bring something from base, only do so with essential items or "big"(expensive) items. Otherwise wait until you have multiple things to bring out. Ferrying 1 ironwood branch at a time to yourself ties up the courier and will make everyone rage at you.
Best Answer
As Dispenser already said: You don't. You can't.
That's the entire point of the exercise, limiting you to 10 minutes forces you to work on what the challenge is for. Giving more time would make the entire challenge pointless, and thus it's not possible.