Starcraft – Since when were carriers and mothership so good

The simple answer is to click on the Nexus. You can see the energy they have. If you have health bars on, you don't even have to click on it at all.

2v2 games are a unique beast in the sense that you need to change your style of play. My 2s partner and I used to die a lot when it came to these early quick game rushes, and on the flip side, have rushed others as well to vary up the play style.

You mentioned that you were able to scout that they are doing a rush, you could mix in your ideas of how to survive to get a good counter to a rush. (Going to be using Monlyth Ridge as the example map and your match up as reference PTvTZ)

At 6 minutes (I assume in game time) they can probably have 10-15 marines and 5-9 roaches.

• If you are confident in your micro, forgo static defenses and simply try to out produce your opponents. At 6 minutes, you can get your defensive 4 gate up. Although it's really close. A mix of stalkers to shoot from the high ground and sentries to hold off the ramp should be able to hold them off. Even a few zealots mixed in there to tank damage while sentries pick away could suffice as well. (So you'd have ~2-3 sentries, 4 stalkers, 1 zealot)
• You shouldn't be afraid to get sentries and use force field. As a protoss player you want to have a fair number of sentries mixed into your army anyway later in the game. The force field only lasts 8 seconds so it is not a huge amount of time to stop you from going to help your ally. If used correctly you may even be able to trap a few of their units in your base (although against a roach/marine army that would just give them high ground vision so not really helping you there).
• You mentioned that your ally is terran, there is very little cost to investing in an early bunker or even two (even with the recent patch, he's only going to lose 25 minerals in the long run if he salvages them). So long as your ally is quick on the repair then he should be able to hold off the rush long enough for you to build up an army and flank with him exiting his bunkers to assist when you believe the two of you have enough to take out the attacking army. (Ally if matching opponent would have 10-15 marines as well maybe a little less because of the bunker)
• I'm not sure how your buildings were laid out, but pylons usually shouldn't be in attackable range, although the new patch doesn't help this point much (which I personally think is a pretty stupid nerf to the race, but I digress). Try to ensure that you have overlap on your production buildings.

Assuming it played out with the above, if the enemy ran to your base first, you force field him off, pick off a few units, or even weaken them as the two of you build up an army to crush the attack since your reinforcements will arrive much sooner than the attacking force. If they attack your ally first they run into a face full of bunkers that are getting repaired and your defensive 4gate should produce enough units to kill that army that they are sending, that or enough to force a retreat.

Going a forge first fast expand with cannons in this case you'd probably need 4 or 5 to just hold off that army if your ally responds quickly enough with reinforcements. Also in this case you are forcing your opponents to either deal with your cannons (with a sentry on the high ground to your ramp to prevent a run by, but if your wall is good enough on the low ground a run by would sustain massive losses) or to attack your ally (who again should have 2 bunkers and since you were attacked first, have ample time to get SCVs down to repair. If you go with the forge first opening I would suggest getting at least 1 sentry and a 2:1 stalker:zealot ratio for the first little while as you'd try to stay close to your cannons when trying to engage.