What is outside diameter (how big around is it)?

I don't have my calipers here at the house, but my steel rule says exactly 1/4"

Is there a plug available for this wire that will grip the cloth cover to keep it from slipping?

Try this: https://www.amazon.com/gp/aw/d/B000FPAN84/ref=mp_s_a_1_6?ie=UTF8&qid=1520707821&sr=8-6&pi=AC_SX236_SY340_FMwebp_QL65&keywords=2+wire+plug&dpPl=1&dpID=41mltD-j76L&ref=plSrch I singe (low heat) the cloth after my wires have been cut to the desired length, and use 1/4 heat shrink in conjunction to prevent fraying. Just make sure your cut lengths are correct or the heat shrink will stick out past the plug.

The cord with which I want to replace has three wires inside, not two. Is this safe to use as a replacement?

If the cord you are replacing is three wire then the appliance requires the third earthing conductor. The earthing conductor needs to bee connected to the earthing pin of the power plug and the other end should be connected to the same point that the old cord was on the product. DO NOT USE TWO CONDUCTOR CORD TO REPLACE THREE CONDUCTOR CORD.

Will this cord work with clip type plug?

I used this wire to make a light fixture out of an old wwgon wheel. These are the sockets I used with it: KINGS E26 E27 Light Sockets, 6 Pack Edison Retro Lamp Holder for DIY Projects 200℃ Heat Resistant No Fire Hazard, E26 E27 Medium Screw Light Holder. You can look at that product to find out if it is a clip type plug. It seems I sliced the cord open, separated the white and black cords and used a screw driver to get the exposed copper under the screw/clip. I tightened it down and put the whole thing back together. The trick I recommend is to use a piece of tape to hold the cloth in place while you are working. It slides up and is impossible to get back down to the end, by the socket. I don't know if that answers the question, but I sure hope it helps.