They are called "wire grommets" or "electrical bulkhead fittings", and if you search for those (or some combination of those words) you should find what you're after.
I typically seal large exterior holes with expanding foam, and smaller holes with silicone caulk. I've run many cat5 and coax cables through walls, and generally keep the hole just slightly larger than the cable, and it seals up fine. You can leave some slack on both sides (I typically leave a drip loop on the outside and a coil of a few inches on the inside where it's not visible), and that's all the flexibility needed. Silicone caulk has enough give that it is fine for this application.
For a larger cable or a couple cables, I'll go through the exterior wall with a conduit, then seal around the conduit with foam and the outside with silicone (expanding foam breaks down under UV, so it's best to cover the exterior bit with some opaque caulk). At the end of the conduit I'll put a wire grommet in a small box (a 90 degree PVC pull box works great).
Outdoor Power - Do not power the camera using an outdoor outlet. You have all the problems you listed and you make the system more susceptible to tampering (an intruder can simply unplug the camera and blind the system).
Powering the camera - Let's begin by checking if the camera and its power supply is rated for outdoor use. The exposure to moisture, and to a lesser degree extreme temperature, makes many indoor units unsuitable. The connection of the power supply to the camera needs to be water resistant. If it is not, you need to consider a different camera unit.
In general, outdoor power lines need special cable or a metal or plastic pipe to protect the cable. The power to the unit is apparently low voltage, so it may not need the same level of protection as it would if it were 120 volts.
You will need to bring the wire through the wall (or through a soffit) at some point. You want to drill through an area that DOES NOT contain framing. You will go through siding, sheathing, probably insulation, and then finish material, probably drywall.
The main issue is sealing the entry holes from water and air infiltration. This can be done on the outside using a good outdoor caulk.
On the inside, you can use a low voltage junction box to protect the wire and then route it to an outlet. The best approach might be to run the wire to a double box, low voltage on one side and line voltage on the other.
You could then have the wire exit the front of the low voltage side and have the transformer plug into the line voltage side.
Mounting Camera - Most modern cameras (at least those without heavy housings or motor mounts) are fairly light, just a few pounds, if that. They can be screwed into the sheaving of the wall, which is located beneath the vinyl siding. You need to drill through the siding and into the sheathing and then screw the base to the sheathing. As you are mounting, you can seal the screw holes and even the base of the camera with outdoor caulk.
If the camera is heavy, you should screw the camera to a framing member (stud). These are found at corners, regularly spaced horizontally (usually every 16 inches from a corner), and near the roof and floor line of each level. You can use a stud finder to locate them.
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Be sure to use a shielded Cat5e or Cat6 cable, as Ubiquiti clearly specifies the use of shielded cable and connectors for outside access points.
Conduit is not required if you use the correct type of cable. Ubiquiti would like to sell you ToughCable and connectors, but IMHO those are overpriced .vs. any other cable that meets the need (exterior and shielded) so my Ubiquiti outdoor APs are connected to another brand that I could get for considerably less cash. It can also be difficult to keep the cable bend radius reasonable if running in 1/2" conduit with sharp corners around the building corner. If you want to use conduit, make the corner with sweeps.
Depending on the building layout and wiring access, running the cable up near the roof (or on the soffit, if there's a soffit) is generally less obtrusive than in the middle of the wall. It's also, not at all incidentally, a drier, more protected location for the hole in the wall (or you can come out the soffit if you have access from in the attic.)
If you are using any of the outdoor APs with the stock omni antennas, your best method is to run direct burial shielded Cat5e or Cat6 to a pole in the yard somewhat away from the house. The APs are well weatherproofed and don't need the shelter of the house, and a pole location in the center of your coverage area will give the best coverage. Conduit with sweeps (not sharp corners) would not be a bad idea if digging that trench, and removes the requirement that the cable be direct-burial rated, though it's usually not an extra cost on the cable. The conduit is a small extra cost in never having to dig the trench again.