'2v2 random' mean that you will randomly be put on a team for each ladder match.
And '2v2 team' is where you are partnered up with someone and you stay their partner throughout the ladder. Most commonly a friend. The only way I believe to set this ladder up, is to have your partner be in your party before starting a 'Quick Match'. You will get placed in different ladders with different friends, meaning you can be placed in several '2v2 team' ladders.
I think both of the answers here are good answers, but since you seem unsatisfied let me try to contribute.
When playing as Protoss or Terran I often find its useful to use my Pylon/Supply Depot to help give me vision around my base. This isn't, by any means, a replacement for scouting, but it does give me a little advanced notice on drops. Since most maps have your main on a cliff, spreading out these (otherwise useless) structures helps monitor not only the edges but extends the range of your viewing.
Now this technique won't stop a drop, and it won't even really give you too much advance warning, but when every second counts, these are invaluable. You also have the benefit of power for Warpgates. If your opponent drops near the edge, these aren't advanced enough warning, but they do provide a nice distraction while you move your army. If he bypasses them and moves to your mineral line, they're a more advanced warning.
By comparison, Overlords almost trivialize this problem as you can spread them about the map, around your base and even around your opponent's base.
I know its not a solution, but it should help.
The truth is, there is no cheap, effective, sure fire way to guarantee you catch drops. Certainly map control helps, as does scouting, but if there was an easy fix for this terran's would probably never drop.
Best Answer
Start by getting your ally's base up to full saturation, and then ignore it and play as if you are on two base yourself. So immediately get your two-base tech and production up.
With the additional income, you will have a window of advantage because you can tech up more quickly than a normal game. To illustrate, for example you're zerg and your ally disconnects right at the beginning of the match. You will be able to build a spawning pool extremely fast with the additional income, making a 6-pool even faster than it already is normally.
So, play an aggressive game, keeping supply counts low and keep your opponents distracted. Do a stim timing, blink timing, etc. Stay in (or close to) battle, where micro is most important. This is your advantage being one player on the same economy as your opponents having to juggle armies. Two players controlling units in a single battle is not as efficient as one because you have a vision of your unit positioning, when to retreat, etc.
You can play on even footing until your supply maxes out. Your opponents can continue producing units while you're capped at 200. So, try not to get to that point and kill them first.