To find outdoor rated cabling, I did a quick DuckDuckGo search for "outdoor rated Ethernet cable" and came up with quite a variety of purchasing and review sites. I suggest you take a look at what's available and make your choice from among those. (Do note, a couple of "review" sites simply list their "top 10" choices with absolutely no analysis of why they've chosen them, so I consider those nothing more than "advertisement sites", YMMV.)
I would recommend going with something that is outdoor rated, as several of the vendor sites note using different jacket materials in order to be UV resistant and waterproof. I believe that gnicko's comment is correct in that all Ethernet cable is made up of solid core wire inside insulation, twisted in pairs, then put in an outer jacket, so I don't believe you have an option to go with stranded. When I've looked at cabling, solid vs stranded was never among the choices I saw listed. It seems I was wrong as Ecnerwal and Fredric Shope point out. According to the page linked by Fredric:
Solid copper conductors are more durable and tough...great for outdoor installations
So that firmly answers the question for "Solid".
Also:
Solid copper conductor Ethernet cable is typically sold in the variety needed for in-wall or plenum (HVAC) space installations, and for outdoor scenarios. The outer jacket type matters a great deal
and is followed by a link to another article at their site describing the outer jackets.
Frame challenge:
Don't go wired from upstairs to downstairs.
I'm a big fan of wired instead of wireless networking (improved speed, reliability and security), but this may be a good use case for WiFi as your main network infrastructure. At a minimum, you might want to look at using WiFi as the bridge to get from the 2nd floor to the 1st floor, then run everything on the 1st floor from a 2nd wired router. If you have this WiFi connection used for nothing but this bridge, then you can pick a very long random password for it for maximum security and only have the inconvenience of having to enter it twice - once in the host router, once in the access point that's picking it up. You might consider wireless cameras, as well, simply for ease of installation.
My router has a main WiFi network and a secondary, "guest" network. I can deny the guest network internet access if I want. I'd suggest a setup like this where your WiFi bridge does not have internet access at all. That would also improve security as someone would have to be inside the wired network to get access to the wireless, and if they managed to hack your password to get on the bridge, they wouldn't have direct access outbound. This is about the extent of my network knowledge, so you'd have to do further research on this on your own.
Another alternative might be to use Ethernet power line adapters. These inject the Ethernet signal into the electrical wiring already in the walls and pick it up from any other outlet in the house. I believe there are suggestions like the best signal strength is from another outlet on the same circuit breaker, so this might make it difficult in your situation to find one breaker with outlets both upstairs and downstairs. You may also run into bandwidth limitations that you find unacceptable. I thought I'd throw this out as an idea to consider, though.