1. Depending on personal preference, money, etc. people typically change strings anywhere from every day to every six months.
1.1. Find out what gauge the strings you have now are and buy several packs of different brands and try them out. Slinkys, Elixers, and D'Addario EXLs are all popular. If you want to try other gauges, you'll need to have your guitar setup adjusted.
2. For your style, I'd recommend a Jet City JCA2112 (combo) or JCA20H (head). They're low cost and suitable for classic rock and blues. Might be a little crunchy if you have to crank the volume all the way, though, and it's Class A/B rather than Class A. I'd say try out a bunch of amps and see what suits you.
2.2. Separate heads and cabinets allow you to mix and match and are easier to load up and haul to gigs. They also seem to be something of an ego boost for many people.
3. I find thin, flexible picks easier for strumming, but it's harder to pick with consistent force. Thicker, stiffer picks tend to make it easier to pick single notes. Pointy picks (e.g., Dunlop Jazz III) are often prefered for pinch harmonics, tremolo picking, etc. Picks are (with a few exceptions) very cheap, so buy a lot of different kinds and try them out.