Outdoor extension cord

Wrap with "self amalgating tape" and then few layers of "Denso Tape". That should do the trick. Use gloves, Denso tape is a bit messy to apply but very effective.

Wrap well in Denso tape as I have advised elsewhere

Well, I think that question has been answered elsewhere with the reference to a commercial item so I'll go with that.

That will clearly depend on how long the run is. However, around 2.5-3mm sq should be ok to about 30ft

link.http://www.taymac.com/taymac/mm2.html Perfect, thanks!

-- You can protect connections from rain and snow (not moisture) by using sandwich bags but you need to check them periodically (as they will deteriorate) and wires from mechanical damage and summer UV (which is apparently not a problem for your purpose) which makes the extension cord brittle and the chance of problems is small. I've done this - the original cord has been replaced because it was there all year round and exposed to sun. The replacement follows a different route so that it is not covered for thermal reasons but is also not exposed to sunlight or mechanical damage.

Note that standard outdoor Christmas lights are more questionable than a good outdoor rated cord and a protected outlet but don't seem to be considered a problem with regard to code regulations.