You bought a reasonable solder. Sn63Pb37 is a (close to) eutectic alloy, so it has a bit more of the more expensive element (tin) and is a bit better than Sn60Pb40 solder. As a result, it melts at a bit lower temperature and (more importantly) it's a bit easier to use because it solidifies suddenly rather than going through a mushy phase where it's easy to create a "cold" solder joint if the parts move during cooling.

0.025" (0.635mm) is a reasonable size for general purpose. I use 0.8mm and 0.38mm.
Personally, I prefer Kester 44 because the "no clean" flux leaves a nasty residue that is very difficult to clean off. For most applications it's just fine, but for sensitive high-impedance analog circuitry it can be a problem. I had an application with 5-50K resistors where it was a problem, because of extreme accuracy requirements. The contract assembler had used no-clean, contrary to instructions. The Rosin RMA flux may look ugly if you don't clean it, but it seems pretty inert, and it's easy to clean with solvents. I doubt you'll see any problems with the one you chose.
You say "solder station". It's temperature controlled? That helps.
You don't need to buy any extra flux for through-hole parts. You might want to buy a flux pen (eg. Kester #186) if you intend soldering surface mount parts- it helps a bit. They look like a felt-tip marker but dispense flux. Unfortunately, they seem to be a bit hard to come buy these days. Bulk flux attracts Hazmat shipping charges.
Make sure you're dealing with "fresh" parts. If they've been laying around in some surplus shop oxidizing for a decade or two, they're not going to be all that easy to solder!
If the tip is well wetted, you should be able to touch it to the two parts at once, feed the solder into that pool, and stop when you have a fully wetted joint. It should be shiny and smooth when it's done. Practice makes perfect.
I suspect if you've had problems with soldering, other than technique, your parts are the source of the difficulty rather than the solder or soldering iron.
Your problem is almost certainly not a direct issue with the solder itself but with the flux in the solder. Many fluxes leave a conductive film on the board which must be removed for circuits such as yours.
Note that RTC chips try to be extremely low power so as to preserve the battery as much as possible. As such, the biasing of the 32.768KHz crystal for the oscillator is designed to be rather low current (i.e. high impedance circuit). The flux conductivity can sometimes be low enough to completely swamp out and literally short out the oscillator preventing it from working.
Best Answer
This sounds like a lot more trouble than it's worth, and probably not even a savings. Heated solder will have partially oxidized and it's flux will be gone. I suppose if you melt it the oxidized part will float to the top and you can add your own flux, but unless you are a large commerical operation using lots of solder I really don't see this being worth it.
I am having a hard time imagining how you'd even collect the used solder in a hobby situation. We have several people here doing electrical engineering and technician work professionally, yet this would make no sense here whatsoever. Used solder would come drops or tiny balls at a time as the irons are tinned or components desoldered from a thru hole board and the holes cleared. If I tried to collect the tiny balls of solder, I'd probably scoop up as much dirt and other stuff like clipped leads as I would solder. A whole role of 60/40 rosin core solder may take a year or more to go thru, so even if all of it could be reclaimed it would be a tiny savings. Just the wasted time dealing with reclaiming solder wouldn't be worth it. And then there are the questions about the resulting quality and the extra energy required to melt it down. No thanks.
Even if you could do all this successfully and somehow profitably, then how would you use the solder? At best you have a small molten pool in a solder pot. That's very different from wire solder with flux core. For the ordinary hobbyist or small scale user, a pool of molten solder is not much use. We don't have a solder pot here, for example. I'd get one if there was a good reason for it, but so far that hasn't come up.
Frankly, this sounds like a really silly idea for small guys like hobbyists and even us.