If I delayed my turn, the 7' tall Orc Fighter in the party picked me up and threw me at the enemy....
...Would I be able to do a full round attack because the ally would be responsible for my movement?
Well, I doubt the standard rules cover this, but this is how it would work under my house rules where 'throw enemy' is a standard combat maneuver and flying charges a fairly common stunt players try (although I've yet to have one actually throw the other one).
Assuming your friend threw you on target and you landed on your feet, then, "Yes.".
First, being thrown any distance causes damage equal to what you would take if you were thrown the maximum distance possible by the throw. So, if your friend hurls you full force, whenever you collide with something you'll take damage equal to the maximum distance that you could have been thrown. So, lets say your friend throws you 60', then regardless of whether the stunt is successful or how far you actually go you'll take damage as if you'd fallen 60'. This damage can of course be mitigated by a successful tumble check.
To throw you accurately, your friend must make a ranged touch attack on the target. If you are being thrown at a square, the target counts as a medium sized immobile object so base DC of the throw is 5. However, thrown shrunken gnome counts as an improvised weapon, and unless your friend has an ability that says otherwise this will mean a -4 penalty on the throw. Additionally, the range increment of a tiny thrown object is 10' for a medium sized creature, so if he were throwing you 60 feet then there would be an additional -10 penalty. So, it follows to throw a gnome accurately at a square 60' away is DC 19 - and the gnome is going to take under most systems 6d6 damage (though see my notes below).
If the throw is inaccurate, it scatters. To resolve point of impact, throw a 1d8 to determine direction and a 1d4-1 to determine the number of squares in that direction. However, the gnome never lands more than 5' behind the thrower and never beyond the maximum possible distance of the throw. If the throw was made at its maximum possible distance, use a 1d3 to determine direction of scatter in the direction of the thrower - the throw is always short in this case! If the to hit roll was a 1, I use a 1d6 to determine distance of scatter but that is very much optional (I also use fumbles in my game, so the players expect it). Note that a scattered throw may still very well leave you next to your intended target to allow for a full attack. A scattered attack may incidentally land in the same square of the target or a different target. I generally allow a DC 20 reflex save to avoid unaimed attacks like this, otherwise you got a lucky (or unlucky hit) and resolved it as aimed at the target (see below).
Once you arrive at your destination you must land or roll to your feet just as if you'd fallen. In my game, if you take no damage from the fall, you are automatically assumed to land on your feat. However, if you do take damage from the fall, rolling to your feet or otherwise fast standing requires a DC 25 Tumble check. If you fail, you are assumed to be prone in the square of your target.
...Would I be able to make a bite/claw/claw attack at the enemy before hitting the ground, or would I bounce off the enemy and fall down?
Depends. If you are thrown at the enemy or land in the enemies square, you resolve this as a Charge Attack. (In my game, leaping or flying charges are also automatically assumed to be made in the Offensive stance.) Thus, you would not be able to make a full attack. However, if the charge attack is successful, your enemy also takes bonus damage equal to the damage you suffer from the fall/impact with the exception that unless you are wearing heavy armor or have a natural AC of 8 or higher, you resolve as impact on a soft surface (-1 damage per die) regardless of whether the target resolves as impact with a soft surface.
If you are thrown to land in a square adjacent to the enemy, you may make your full attack as normal upon successfully landing.
However note that in either case, because you are tiny and therefore have no reach, you draw an attack of opportunity from your target.
...What would the range of throwing a tiny ally be?
Maximum range of an improvised thrown object is equal to the throwers reach + 5' times the maximum lift of the thrower divided by the weight of the object times 4. Objects weighing less than 1lb are assumed to weigh 1lb. This is a rather complicated formula I admit, but it produces pretty good results. A normal human can throw your 5lb gnome about 30 feet. If your friend has 18 strength, he can throw your 5lb gnome about 80 feet (5 + 5 * (300 / (5 * 4)). He can throw 50' stone about 10 feet, and a 1lb rock or a ball about 375 feet. That's not perfectly realistic, but its close enough for the purposes. Note also that I use special falling rules where objects of different sizes take different amounts of damage for falling - a tiny object for these purposes would take -2 damage on the dice (minimum zero). This revises the standard damage for falling 60' from 6d6 to 6d6-12 (avg. 10). This is important because if you are going to have the target take damage from the impact equal to the damage taken by the thrown object, you want to avoid the unrealism of having improvised throw weapons hit with the force of cannon shot unless they really were thrown with the force of a cannon shot. If you don't, you'll have players carrying sacks of rocks and beaning things as a primary strategy.
Alternately, you can compute maximum thrown range using standard range increments, but this will create problems for throwing say sacks of grain, chests, or medium size allies. But it should work ok for objects of the 'normal' size people throw (under 20lbs).
Also, can a sling launch a bullet weighting 4.375 lbs? If not a sling, any other weapon?
Yes, however it counts as throwing an improvised weapon and uses the above rules, since the sling wasn't really intended to hurl such a heavy stone. If the wielder was proficient with the sling (an exotic weapon under my rules), I'd give them a +2 bonus to effective strength for trying a stunt like this.
