Hi Here. It sounds like you are currently doing partial boils (3-4 gallons) in your five gallon pot. If you are going to continue to do that, I don't think you probably need an immersion chiller - an ice bath and cool top-off water probably cools your wort down pretty quickly, at least that was my experience.
The chiller is really necessary when you do full boils (6.5 gallons for a five gallon batch, including boil-off). I think the chiller you linked to looks fine. Its not huge and won't do the job super quick, but it will do the job just fine.
The stove looks fine, i know others here have used it with good success. I use a similar stove by the same company - the Bayou SQ-14 http://www.amazon.com/Bayou-Classic-...4584650&sr=1-2 I like it because it has a big base and holds my brew kettle securely. I've had good luck with it, fwiw.
Again, if you are doing partial boils, do you need a big propane stove? When I did those, my regular old kitchen stove worked just fine with less hassle.
As for a bottling bucket, I think using one with a spigot (that you attach a hose and bottling wand to) is a good investment. Having the right equipment (spring-loaded bottling wand, vinator, bottle tree) makes bottling go much easier and hassle-free.
Hope this is helpful!
Edit: I didn't notice your last question - yes, in fact you should let your priming solution cool down to room temperature before adding it to the beer.