You can try, good luck with it though.  I've only had a couple of these in recent years that I've been able to repair on the outside of the joint.  Reason why is because of the dynamic pressure forces that are present in spa plumbing, and the ability of this force to push outward, against any kind of patch or seal.  Since the seal only has something to attach to, and nothing to help it resist the force, then the water usually wins.

That being said, what I would try to do, first, and depending on how accessible the leak point is, with the tub water level below the jet, and the PVC joint itself totally dry, I'd sand down and roughen the surface... all exposed surfaces around where the leak is.  Then, I'd take some heavy bodied pvc cement, (oatey green or blue can), and coat the entire area.  I'd let it dry for about an hour and then do it again.  Once the glue has dried (the second time) for at least an hour, then, I'd use some of the hand mixed fix-all epoxy putty that is available at the hardware or automotive parts stores.  This stuff comes in a tube, and you cut it, and then hand mix it until it becomes a solid color consistency.  Then I'd apply the epoxy putty all the way around the joint, and overlap the pvc cement by at least 1/2".  I'd next keep my fingers crossed, and wait until the next day to see if it worked.

The reason why I use heavy pvc cement, is to create a small gap-filler to actually stop the leak.  The epoxy putty, provides some 'back-bone', to prevent the water pressure from pushing the pvc cement out of the gap.  Both of these are necessary, as the PVC cement isn't strong enough to stop the leak on its own, and the epoxy putty isn't thin enough to fill the micro size gap.  Both together, will sometimes work.   I've tried using silicone sealant in this same application, but because it flexes outward, even with the epoxy putty over it, (it will compact under water pressure), it generally will still leak.

Usually not.  Most jets are usually sealed with silicone sealant between the jets and the wall of the tub, on the front and back.  Over time, if the jets get moved in the socket, the hoses jostled, or sometimes, just sitting idle with no water pressure in the tub itself, you will find that the joints will begin to leak a little bit.  If the leaks are small enough, what I usually prefer to do is to leave them alone, and hopefully, small particulate matter in the water, will eventually collect in these micro-fine gaps, and stop the leak.  Also, you may notice quite a difference once the water gets hot again, causing the jet/sealant to expand a bit, and that can stop the leak too.

I always prefer to give these small leaks a lot of time to stop on their own before taking such drastic action of cutting out the jet and patching the plumbing.

If it doesn't stop after a week or so, then I would run a small bead of 100 percent silicone sealant around jet INSIDE the tub, between the surface of the tub, and the outer ring of the jet.  Then using my finger, smooth it out so that it's barely noticeable, let it cure for 24 hours, and then refill the spa.  That usually solves all of these kinds of leaking jet problems.  But don't try this on the back of the jet, you'll be wasting silicone, as the water will push the silicone right off of the joint.  With the silicone inside the tub, the water pressure will push the silicone INTO the joint, creating an even better seal.

You bet it is.  If your pump motor bearings haven't been toasted yet, then you need to do something about your water quality.  Your pump seals are failing, and if your water quality continues on the present course, you will be in for a big suprise shortly down the road.  Check out the water quality section here.  If you've corrected the water quality and still have the leak 2 weeks later, then you need to disassemble the pump wet end, and replace the seals right away.

There's a couple of options here.....

1.  Replace the gasket around the canister top... (or bottom).
2.  Put some lubricant such as "Water Lube" around your gasket. (The lubricant acts as a gap-filler).  If you need this problem solved right now, then I've resorted to non-hardening pipe sealant such as T-2, (with Teflon) by Rectorseal.  It's a little messy, but it does work.  It doesn't harden or mess up your water.  T-2 (or other comparable non-hardening pipe sealant) is available at most hardware and home supply stores.

Generally, yes, you will have to replace it since this type of filter is installed as a 'pressure side' filter, subjected to between 10-30 psi.  Quite a lot of water pressure for a patched crack to survive; and $200 is not a bad deal for this much labor and parts to replace a mounted filter cannister; (275 is probably about par for the course).   I'll tell you... these things are not that easy to replace, and in some installations, you may have to replace it using a miniature spa tech about the size of your tool box to crawl under there and do the plumbing work for you!  All kidding aside, it isn't the piece of cake that it may look like, especially with the limited lengths of plumbing, and the fact that you may have to flex or bend the existing plumbing just to get it to fit properly, and still have enough patched pipe in both sides of the filter assembly, deep enough, to provide an adequate seal.  Then with all of that done, keep your fingers crossed that the plumbing that you've disturbed didn't cause leaks at the components that they were connected to.

In most cases (note that I said "most"), you don't.   Water feeding into the blower can be cause by a number of factors, and depending on how the blower loop (the loop that rises above the waterline to prevent this) is designed, you could probably consider it a freak occurence.
There are several things that can be done to prevent a destroyed blower again.

1.  Be sure your spa is connected to a GFCI.  If even the slightest bit of water creeps into the blower, it'll trip that thing in a second, and you should be able to figure out what's happening before the water has a chance to really damage the motor again.  (Let the blower sit disconnected for a few hours and let it dry out, or use a blow dryer to dry it out).

2.  Watch your water level!  Be sure that you keep your water level no higher than 1/2 way up the skimmer.  If you load up the spa with a lot of people, and water starts spilling over the side, then there's probably a good chance that you may flood the blower loop.  The only thing beyond this to protect the blower, is a one way spring loaded valve.  If that fails, then water will get into it.

3.  Install a 2 inch PVC "T" at the blower port, with the bottom of the T, plugged with a PVC plug, pointing downward, with a small 1/8" hole drilled into the plug.  Any water that gets past the loop and the check valve, will usually seep out the hole, or get blown out of it when the blower turns on.

Small leaks can usually be found with a close detailed inspection of the spa equipment and the plumbing underneath, but if you've reached your wits end, then here's some suggestions that may help. 

Pressure Switch, or Flow-Switch.  This device protects the heater from burning itself out.  All pressure and flow switches will eventually fail.  Slow, creeping leaks are quite common.

Filter gaskets.

Heater Element attachment bolts.

Spa drain fittings and pipes.  Particularly if the spa was not completely drained during a freeze, remnants of water will freeze and expand, stretching the limit of the pvc fittings, pipe, and sealant. 

Cracked Light Fixtures.  Doesn't happen often, but it's an excellent candidate.

Shut-off Valves

Filter housings and pressurized filter covers.  This can really be a tricky one.  You've got to look real close for some of these hairline cracks.

Jets and plumbing.  Possible, but very unlikely, unless subjected to freezing temperatures with water in the pipes.

If the spa has been moved recently from one home to another, you increase your chances of getting a small leak by about 40-60 percent.   Many times, if it's a very slow drip leak, there's a good chance that the sealed fitting will reseat and stop leaking with a little time.  In this case, my best advice is to let it drip for a week or two just to see if it will.

If you own a spa that has full foam insulation, and your spa is seeping water from the bottom, then you may have quite a chore ahead of you to find it.  Particularly in the case of a spa that has been moved, the first place to check is the bottom to see if you can find any places where something may have entered into the bottom of the spa and struck one of the pipes, jets, or drain fittings.