There are two capacities for rear derailleurs.

1. Largest cog it can handle. This has to do with the upper pulley keeping off the cog. There is some adjustability here (B-Screw) but off the top of my head, I think the Shimano road derailleurs can handle a 27 tooth cog.

2. Largest difference between smallest combination and smallest combination. If the chain is long enough to fit the largest-cog/largest-chainring, will the derailleur be able to pull all the slack out of the smallest-cog/smallest-chainring combination. If you make the chain too short, trying to shift past its capacity (as an example,. the large/large) will break something, usually the derailleur. If you make the chain too long, the chain will droop in the small/small combination. Subtract the small+small tooth count from the large+large tooth count to get your difference.

I think the derailleur capacities are stamped on the derailleur somewhere.

I would say a mountain bike derailleur would be needed to handle the 30+ tooth cog.

The shifters used make no difference on the derailleur capacity.