That's your yoke spring. It lives under the the horizontal bar (yoke) that moves the sliding pinion on the stem. It's upside down in the picture, just flip it over, and carefully put it with it's long leg against the underside of the yoke. Be warned: This may sound simple, but these springs will go into orbit, so a good pair of tweezers is essential. I hold the curved end down under a chisel shaped end of a pegwood, and carefully place the legs in position, one at a time. The spring will of course be under tension, so make sure it is laying flat in its cut out area on the plate, with the curved end left, against the curved cutout on the plate.
As long as the jewel is the exact same size (hole size, diameter, etc) as the one you're replacing, and is friction fit, you can do it with a Seitz tool. You also might be able to do it with a staking set, but I wouldn't do it unless you had either of these, and they are expensive tools. You can't glue jewels in place, for example, they have to be exactly placed. If it is not a friction fit, ie., is an old pocket watch, it is not a simple job. This is something I would not recommend trying, in either case. Jewels have to be placed and adjusted for the end shake of the pivots they are for. You could do the prep and bring both plates or bridges or what have you into a watchmaker's shop, and tell him exactly what it is you are trying to do. He will have the gauges necessary to measure the pivot holes and jewel sizes, and may even have loose jewels in the size you need. Again, the newly placed jewel will have to be adjusted for end shake which is checked when the pivot is in the hole and the bridge assembled back into the watch.
Could be any number of things. Did the watch run recently, or have you just acquired it and it has an unknown history? Pocket watches are especially susceptible to shock. If they are dropped, the balance staff usually breaks. If your balance wheel wobbles when you carefully touch it with a toothpick (don't touch the hairspring), the staff is broken. If the balance does not wobble, but will move easily back and forth on its axis, your staff is ok, and you have other issues. The watch probably needs a service.
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