making a twisted up, sometimes useless piece of wire into a straight piece again.
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The most significant part of the equation is, what will be the maximum current draw? If it's to be a small current draw, for example 40 Amp service (40 amp breaker), you'll use 8AWG wire that's rated for direct burial. Higher current rating will require larger wire. As always, if you have any questions about the proper wiring or how to connect electrical service, contact a local electrical contractor. Understand that improper wiring can result in fire and/or death.
This can not be answered with the information you have given. 60Hz does not relate the the current draw so wire size can not be calculated.
Yes, this is a safe connection. A range is usually wire rated at 40 amps and the manufactures do not make a 40 amp rated receptacle. The code requires the next highest rated receptacle be used which is a 50 amp rating. This is why the range receptacle is rated at 50 amps. This receptacle is known as a 3 pole 4 wire grounding receptacle, 14-50R 125/250 volt. Black wire to terminal X, red wire to terminal Y, white wire to terminal W and ground wire to terminal G.
Yes, the red wire is a positive wire (+) and the black wire is a negative wire (-) :D
Yes. I am assuming this is a residential question. A better answer is...if the 8 recessed lights draw fewer than 16 amps together, it is good.