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Is it possible to shorten or extend a chain only one link as shown in the photo (in red square). Or do I have to shorten at least two links in order for the other end to match?
The reason I'm asking is that with my current setup (front 52/36, rear 32/11) if I remove one link the chain becomes too short to shift to 52 front and 32 rear. If I extend the chain one link and shift to 36 front and 11 rear, the chain rubs on the derailleur.
Simple answer is "yes". And as it was said already, you need a half-link to be able to connect the chain again.
The question is - why do you need it for? There are couple of solutions to handle/overcome this issue if your dropouts don't allow it, such as:
The OP has shared some information why the question was asked in the first place.
If I understand correctly, the current derailleur is Tiagra RD-4700 SS, which according to the specs I've found is short cage model:
The OP is asking if switching to Tiagra RD-4700 GS (long cage model, according to what I've found) would be the solution to the problem:
I'd say: yes, using a longer cage model would solve the problem definitely since the longer cage allows for a longer chain (serving the largest chainrings and sprockets) and takes up more slack of the chain (serving the smallest chainrings and sprockets).
I don't own the pictures presented in this post, I've taken them from "the internet" and those particular were found on a ctbike.pl webpage, a bicycle parts retailer I'm not affiliated anyhow, just found them using a well-known search engine.
For 1/8" chains as used on singlespeed bikes (or with hub gears), half links are reasonably common.
They do exist for 3/32" chains (which are sometimes used without derailleurs) but are much less common as single speed 3/32" chains are. They're not needed if you've got a test derailleur as the derailleur provides much more chain length adjustment. A chain tensioner can do this on a singlespeed.
The main reason for using these appears to be on singlespeed conversions, when they lack the horizontal dropouts of a frame built for single speed.
Unless using a special half link (as mentioned and linked to by Chris H) to shorten a chain you have to remove the same number of outer and inner plate links. The distance between the joining pins is 0.5 inch so joined chain are always a multiple of whole inches long.
This raises the question of why you want to reduce the chain length by less then one inch. Chain sizing and tensioning for both derailleur equipped and single speed or hub geared bicycles can be accomplished without the need to shorten a chain by 0.5 inch increments.
Update based on new info.This is in agreement with what @Mike said but with specs and numbers.
Derailleur total capacity needed is (52-36)+(32-11) = 37
Tiagra long cage GS derailleur has total capacity of 41. The short cage SS 33. SS model can't take up the slack and is bottoming out the A pivot (what the cage rotates around). Going to a GS model will fix the problem.
Conventional wisdom is to avoid the big/big and small/small combinations, because of the extreme chain angles. This might well be outdated given more laterally-flexible modern chains, and I certainly do it all the time. But you don't need to, since you have ratios equal or close to the big/big and small/small combinations on the other ring - 52/36 is (almost exactly) 36/25, and 36/11 is (approx) 52/16. So use those, or close equivalents, if you want the gear ratios offered by the big/big and small/small combinations. Go with the longer chain so you can shift into big/big without anything grinding to a halt.
