If you're going for leaded solder, I'd go for 63/37 Sn/Pb as that alloy is eutectic, i.e., goes straight from liquid to solid. 60/40 Sn/Pb is another very common alloy. It has a small (5ºC) plastic region, so it transitions from liquid to 'goo' to solid. If you disturb the solder joint as it transitions through that plastic region, you'll have a cold solder joint.
There's no real point to using "silver solder" (62/36/2 Sn/Pb/Ag) unless you're soldering on silver clad connectors, such as many RF connectors. Sn/Pb solders dissolves the silver, which can result in bad solder joints. Adding a bit of silver (usually 2%) to the solder solves that. That said, I also don't think there's any harm in using the "silver solder" everywhere, aside from a possible greater hit to the pocket book.
I disagree that lead-free solder is out of its element in hand soldering, though I can certainly see how someone would arrive at that conclusion. One of the first lead-free alloys was SAC305 (96.5% Sn, 3% Ag, 0.5% Cu). That alloy is still used quite a bit for surface mounted parts populated by machine and you can get it in the form of wire solder as well. It's an awful alloy for hand-soldering. It has a very large plastic region and a high melting point. It doesn't flow nicely and requires a significant dwell time to wet properly. On top of that even a good solder joint made with SAC305 looks grainy (so it basically looks like a cold solder joint performed with 60/40), which makes visual inspection of solder joints harder.
There are more user-friendly lead-free alloys, however. I've come to really like the 99.3/0.7 (Sn/Cu) alloy. It's available under various trade names, such as Kester K100LD, AIM Sn100C, and a few others. This alloy is eutectic and performs a lot like 63/37 Sn/Pb leaded solder in my experience. The only drawback is the higher melting point of 227 ºC vs 181 ºC for 63/37 Sn/Pb. You can get germanium and bismuth doped versions of this alloy as well. Supposedly the Ge/Bi dopants help increase the surface tension so you get fewer solder bridges and such.
For those interested in more details on the various flux types and solder alloys, I do write quite a bit about it here:
https://neurochrome.com/pages/choosing-solder
Breathing the flux fumes, although tempting, is generally not recommended. I use a Weller WSA350 smoke absorber to remove the flux fumes. It's basically a fan with an active charcoal filter. It's pricey, big, and loud but it does the job very well. One could get the job done with less though. An old PC fan with a charcoal filter could work well enough.
Tom