1) The cheaper way to reduce summertime attic heat is to add more passive ventilation. Ridge vents and "whirlybirds" are easier to install, as they require no electrical connections. Also, take a look at your intake ventilation - which is usually in the soffits. Many houses are built with insufficient air flow here, or insulation has blown over the tops of the vents and obstructed them. I would definitely add more cheap and easy passive vents before getting a powered roof vent. I'd also probably pay a little extra for a solar-powered roof vent rather than wiring one to the house. Finally, if you are within a few years of getting a new roof, maybe try a light-color material at no extra cost and see how much that alone reduces attic temps. It's a dramatic difference.
2) Possibly, but only if you have insufficient ventilation as described above, or excessive attic heat (greater than maybe 120F on the hottest day of the year).
3) You may need to have an electrician assess how many plugs or lights are on that bedroom circuit. The maximum load of all appliances that could possibly be run at the same time is only 80% of the breaker's load (probably 20A x 0.8 = 16A). Each outlet and your light fixture counts toward that load and usually 6-8 plugs or lights max it out. There's a decent chance you cannot add to that circuit, because the cheapest way to wire a house is to max out every single breaker. That said, home inspectors rarely do this time-consuming testing so you would probably get away with it. The downside would be if you had breakers flip when running a bunch of things plus that fan at the same time.
4) There's probably a service box somewhere in the attic you could tie into. The ceiling light box will probably only have electricity when the light switch is on. It would take trial and error to figure out which breaker supplies which box. Some houses (like mine) have everything except the overhead lights wired from the crawl space, so finding a good place to tie in might be challenging.
ETA: Regarding whole-house or attic fans, these only work when you want to exchange inside air for outside air. I.e. the outside air is cooler and there's not too much dust or pollen. First, how often does this scenario really happen? Second, simply opening windows and/or doors will accomplish the same thing, just more slowly. Also I'm wary of any solution that adds penetrations through the insulated envelope of the house, because those penetrations dearly will cost you in the wintertime.
