1. Electrolytic and paper capacitors. Anything with wax on it, or that black plastic one in your picture. The electrolytics contain a liquid which is either dried out or almost dried out by now, while the paper dielectric has absorbed moisture and become leaky. Ceramics and Mica will be fine. Paper can be replaced with superior poly propylene or polyester.
2. No, but the few survivors won't have much juice left in them. It's a choice. Know by checking the ESR (equivalent series resistance) or dissipation factor. You can get dedicated ESR meters or use the ESR function on a LCR meter like a DE-5000. Capacitance meters are kind of useless here: the measured capacitance tends to increase as they dry out.
3. Not close at all. They can be reasonably bigger or higher voltage. Note that a typical tolerance on an electrolytic capacitor is +80%/-20%. A 0.01uF is not going to be an electrolytic though.
4. Depends on your restoration style. Some people gut the cans and stuff them full of new caps, then carefully reassemble them. It helps that new caps are quite a bit smaller. I'm more of a resto-mod guy, so I mount a bunch of new caps to a piece of perfboard cut to the footprint of the can, and replace the can with that. I think the most sensible method is to cut out the can but leave it in place, and mount the new caps under the chassis.
And 5:
5. Those carbon composition resistors drift badly. Some people will replace them wholesale, but I just test and replace the bad ones. Either way you'll want to address them. Almost all the time you can replace them with a metal film capacitor, but beware of the standard carbon film resistor since they do not withstand high voltages and tend to have high inductance making them unsuitable for high frequencies. Metal film resistors can be unsuitable for high voltages too so make sure you check the datasheet before ordering.
Check out Jordan Rubin's restore of a Heathkit O-12 for some ideas:
or Paul Carlson's O-11: