First, there is NO Lead in flux. Lead can be found in a "Tin-Lead" solder, but tin-lead solder is getting harder to find. More and more they are getting away from lead. Any solder marked with ROHS (Removal Of Hazardous Substances) means a solder contains no lead. It's become an environmental issue.
Second, lead does not cause cancer, it causes brain damage. So don't eat the stuff.
Third, soldering outside in the cold - the cold will not affect the electronics adversely, however static electricity (referred to as ESD - or Electro-Static Damage) can. The biggest problem with soldering in cold environments is the solder iron won't be able to maintain temperature unless it's one of those with temperature controlled heating elements. AND the cold board will be more difficult to solder, so don't set up in the garage. Don't solder on vinyl or plastic work surfaces. Avoid cardboard and paper. ESPECIALLY avoid styrofoam. TONS of ESD there. Don't use tapes - especially scotch tape (the clear stuff). Masking tape is not good but not terrible.
As for fumes from soldering - you're right, you don't want to be directly breathing the fumes and smoke from the soldering process, however it won't kill you. A small muffin fan set near the work piece, blowing away from you will tend to draw the smoke away from you instead of allowing it to rise in your face. Knowing this - even I don't have a fan. Why? Just because. No special reason, just that I haven't gotten around to doing that yet. One day I probably will, but generally, soldering in my basement workshop - hasn't killed me yet. And my brain still farts normally - I mean "Function" normally.
As for the final part of your question - no, smoke from soldering won't set off the smoke detectors. Not unless you decide to melt a whole spool of solder all at once. You mentioned latex gloves. Not necessary. Can produce some ESD, but I wouldn't expect a lot. The WORST thing you can do is sit on a rug with a plastic rug protector and sit there in socks. You'll be generating over 20,000 volts of static electricity. You WILL be blowing out your electronic devices long before you even power them up.
Research ESD before you start. It's worth understanding, and knowing what to watch out for. I've been playing with electronics for many years. I have a work surface on my workbench that aids in preventing ESD from having any negative effects on my electronic circuits. The paper mask is probably not necessary. You're not going into surgery.