splicingtransformer
I am installing a doorbell transformer for a new Ring doorbell and I have an outlet right next to the main box. It would be simple for me to take the existing wiring, branch it and use it for the new transformer (which I could attach to the side of this junction). That would save me from having to run any more wire – simply take the wiring provided as part of the transformer and splice it to the existing run. Is there any reason that this is not a valid setup before I go about doing it?
Since these are NM-type cables we're dealing with here, the ground wires in the box will be bare wires, and lo and behold, there's a whole bundle of them improperly spliced together in the bottom of the box. Take a suitably sized wirenut and nut the green wire on your transformer to that bundle of bare wires (this will also properly splice the bare grounds to each other).
The black wires in this box are all hot, and the white wires are all neutral (otherwise, there'd be a white spliced to something that isn't white, that's a clue that shenanigans are afoot). So, in this case, it's black nutted to black, white nutted with whites (and you may wish to replace the splice of the existing neutrals with a proper one while you're at it), and grounds as above.
Once you remove the cover of that box with the switch and everything else, you should see three wires coming into the top of the box from the transformer. These are usually white (neutral), black (hot), and green (ground). Take a picture of all the wire connections and make sure that everything gets connected the same when you wire in the new transformer. There will be additional wires in this box that are connected, and these connections all need to remain the same.
The transformer will most likely be physically attached to the top of the box with a conduit nut (inset of picture).
These can be hard to remove and install in a small box, but the basic method is to place a flat headed screw driver on one of the raised portions of the nut and tap it with a mallet to spin the nut in the correct direction. The new transformer needs to be tight. Loose connections lead to loose or damaged wires, and that leads to bigger issues.
When you reattach the low voltage wires to the transformer (the red and white in your pictures), it doesn't matter which screws the wires go to because it is alternating current (there is not a positive side and negative side like with a DC power adapter).
Any "modern" doorbell transformer will be 16VAC, which is what the Ring wants. You are correct that this 10VAC transformer is not powerful enough, regardless of how you wire the Ring. Note that if you plan on using your current chime, a higher voltage transformer might not work. It could damage the chime (immediately, or in the long run). Plan on replacing or removing the chime as well.
Best Answer
That should be fine, it looks like that's just a general purpose receptacle and it's OK to add a small load like a doorbell transformer to that circuit. It looks like there's room in the box, and clearance so the transformer will fit.