Your Shimano 105 rear derailleur possesses the standard (well, since they came out with it back then until only recently) 1.7:1 cable pull ratio. Basically, outside of 7400-era Dura-Ace, 10-speed Tiagra (4700), and all the 11-speed stuff, any Shimano SIS road rear derailleur will work with your shifters to shift those 6 gears out back. There are about eleventy-billion of them to choose from, thankfully. Any Shimano road RD of those choices will handle a 28T no problem, even if Shimano shorts their capability by 2 teeth.

Kudos for you getting up that climb. Looks like you have Biopace chain rings. Do you like them? If not, getting some normal round ones can help, though Shimano designed that 42T inner ring to behave like a smaller chain ring and out of the saddle in grind mode, the Biopace stuff worked well for me. For anything normal, roundy-round rings are the only way for me.

Do you have an approximate budget in mind for these several upgrades? That drives a lot of things. 6500 brakes are very nice, but I personally would bump to the next generation of Shimano road stuff (105 - 5600, Ultegra - 6600, Dura-Ace - 7800) as the geometry got better, dealing with wider rims and tires more easily, and being stronger overall. Most anything in the dual pivot architecture will be an upgrade over what you have now (single pivots), and getting good Kool Stop pads will help noticeably. So if you're set on 6500s, then go for them.