Waterproof enclosure from Amazon with cable glands on everything and hope for the best. A good power supply (Meanwell or almost any knockoff) will have a thermal cut-out that will kick in before the box melts, but oversize it a by a few inches. At -30 / sub-freezing the rest of the time you won't have an issue.

I'd add a temp sensor because I rarely even trust myself. But the spec sheet should provide some additional comfort, the last pages are usually graphs of permissible load for a given temperature.

Not a chance, power supplies contain capacitors that don't do well at lower temperatures. Even special kinds don't survive long at those temperatures. You would need to build an isolated enclosure and include a heater and heat management system. This is usually done with multiple layers, so the inside where the electronics is can be maintained at a constant uniform temperature. Moisture is also an issue, so you'd have to deal with that. All that takes up a lot of power, which will cut into the batteries.

Batteries are a bad choice for something like that. Led lights are super efficient, but when we're talking about something like a couple of strings around a house it's a lot of power. Everyone who has to keep their car (or if you don't have a huge garage their car battery) inside during winter knows that kind of cold kills batteries. So I would recommend powering it from the grid. Something like an outdoor outlet would do fine.

Much easier would be to just have everything inside and only have a single cable with 24 - 48 volt (depending on the setup) for power and some (or one) control wires. These kinds of wires aren't that expensive and easy to run. With good water proof connectors it's easy to plug in and wire up. They aren't very thick, so only a small hole is needed. Usually they connect to wire terminals on the inside, so the hole only has to be big enough for the wire.

I think you can get kits for this kind of thing, which makes it easy to setup and get running. Just a pain in the neck to step on a ladder and securely mount it to the roof.