(1) You DO NOT need a capacitor. Ever! Don't waste your money. It cannot prevent an amplifier from overheating. It is also of extremely limited use for other reasons.
(2) Bridging means two channels are tied together to make one single channel. The power is equal to the power of each channel added together. The impedance will be twice the impedance of a single channel. Therefore, if your "350 watt" two channel amplifier putts out 175 watts RMS per channel at 2 ohms, in bridged mode it will put out 350 watts RMS at 4 ohms. The main reason to bridge the channels is usually because a sub needs more power than a single channel can provide.
(3) A common reason for an amplifier quickly overheating and cutting out (going into protection mode) is that the attached speaker or speakers are lower in impedance than the amp is rated for. So, if that subwoofer is 1 or 2 ohms, and the bridged impedance of the amp is 4 ohms, the amp will overheat. Without a thermal protection circuit it would literally burn up the circuitry.
(4) Either you completely misunderstood what the installer was trying to explain, or that person doesn't know what they are doing. My guess is the installer is either not really knowledgeable OR he's trying to earn a few extra bucks.
(5) It would be a lot better if you told us exactly the brand and model numbers of the amplifier, the subwoofer and the door speakers so I can verify if the problem is an impedance mismatch or something else. Also mention whether the Kicker door speakers are being run off the amplifier, or directly off the Radio / CD player. You cannot adequately run 4 speakers and a sub off most 2-channel amplifiers.
(6) A distribution block isn't expensive but it has nothing to do with an amp overheating. A distribution block is most often used in a 2 amp setup so that only one long power cable needs to be run from the battery. I'm guessing he's using it to add the capacitor to the system which is completely unnecessary. No capacitor needed so no distribution block needed. It's that simple. You installer is barking up the wrong tree there.