I am somewhat new to working on pinball, but have been working on other stuff for over 20 years (vintage hi-fi gear, antique radios, guitar amps, microcontroller DIY projects, etc).

Through trial and error, I've settled on Kester #285 63/37 almost exclusively for point-to-point and PCB (occasionally use #44 for really dirty connections, but I am not entirely sure this is absolutely necessary). #285 is mildly activated rosin (RMA), which is less aggressive than the activated rosin (RA) in #44, but to me there is little difference in ease of wetting and flowing the connections. 285 residue must be cleaned, but is much easier than 44 to do so thoroughly with IPA alone (91% or greater only) and wets and flows a little easier than #245, which is a "no-clean" flux and is less aggressive than both 285 and 44 (more aggressive = more acidic/stronger reducing agent). For stubborn residue, I have always gotten good results with Techspray Flux Remover, which is a stronger solvent than IPA alone. I have noticed that #44 produces more copious smoke than other rosins, which may be what the OP is seeing. I use a small fan to blow the smoke away from the work area, as rosin smoke is a fairly severe eye irritant (the proteins which comprise your corneas do not appreciate having their oxidation states screwed with!)

I really don't like using soldering guns for much of anything electronic because they provide way more brute force than necessary. Decent temperature-controlled, low voltage soldering stations can be bought for around $100 and are much easier to use (in my opinion) and much less likely to overheat/burn stuff; compared with thousands of dollars worth of pinball machines, it's a tiny investment which will pay off big time.

I don't think leaded solder is going away any time soon. Unless things have been amended in the past couple of years, leaded solder is still legal under ROHS for repair purposes and prototyping, and there are some exemptions to the lead-free rule for certain classes of new equipment as well (avionics and medical devices for example).

I typically start at 300C (~575F) and increase as needed to a max of 330C (~625F). (Lower temperatures are more risky for damage to parts/boards than higher temps due to the longer contact time required to get a good connection.)