A "slightly bulging" capacitor might still work but it should be replaced. 
(But if you are troubleshooting a circuit board this might not be the only problem.)
Good! They do have to be out of the circuit to measure correctly.
In fact... Once you've removed a component it's almost always better to replace it with a new one unless you are 100% sure it's good, or unless you've already found the "real problem" is something else. And if your goal is troubleshooting/repair it's almost never worth setting-up a "special experiment" to test a component. (It's OK to do a quick-check you can with a meter or with whatever you have handy.)
There are some multimeters that can measure capacitance.
Right. When voltage is applied (from the Ohmmeter) current flows and it will read some resistance. The current decreases as the capacitor charges-up and when it's fully-charged current stops flowing (except for "leakage") and you should read very-high or infinite resistance.
The charge timing depends on the capacitor value and the meter. If you have a known-good capacitor of the same value you can compare. Otherwise, checking with an Ohmmeter is a very-rough check.
Of course if the capacitor is shorted (nearly zero Ohms) or if it reads "permanently" low resistance, it's bad.
Reverse operation of a polarized capacitor is "undefined".
If you suspect a capacitor on a board, a quick-and-easy check is to stick another capacitor across it in parallel. (Usually you can just hold it place without soldering.) In most applications a higher-value cap is OK so it doesn't have to be identical. If the capacitor is open or has simply "lost it's capacitance" the circuit should start working. Of course if it's shorted, this won't work.
The only type of capacitors that commonly fail are electrolytics and tantalums (polarized capacitors).
Ceramic capacitors, film capacitors, etc., almost never fail in "normal operation". Of course you can kill one with excess voltage but that's usually the result of a bad/failed experiment or a bad design and you virtually never see it on a good-working board. These small-value capacitors usually charge quickly so they normally just read open (infinite resistance). The leakage on these is usually super-low so they usually do measure open.