I am trying to combine my clock cord and my lamp cord into one so that I only have to plug one thing into the power outlet. Would it be OK to splice them to a single plug?
Electrically, no problem: splice away. Both clock and lamp will work fine.
Safety-wise, you really shouldn't do that. It would be easy to end up with a cord that could be pulled apart, leaving house voltage exposed and ready to make your life overly interesting. And, the plugs may be polarized to ensure that the "hot" side of the outlet connects to the right part of the lamp or clock; if so, it would be easy for you to get this flipped around, leaving the lamp socket invisibly energized even when its switch was turned off.
It would be far better to buy an outlet tap or strip so you have more places to plug your lamp, clock and other items into.
"Safety" here will be the key of your work. I assume, that you do not intend to use any of the other possibilities, and combining the power cords is the only option you pursue. If possible, make a photo of each device and how they are located, this might help to improve my tips, and here they go:
- Since you are asking such a question, I assume that you are not very experienced. Thus: before starting any modding with electrical devices: tell somebody that you are going to do it, and where. And ask them to check on you every few minutes.
- Don't use (electric) tapes. They are ugly, and only good for temporary solutions. Use shrink tubing instead.
- "Twisting" of wires is only good as a temporary solution. Solder them instead.
- If you don't know how shrink tubing or soldering works: leave it to somebody else.
- Use a safe environment with circuit breakers for testing your mod, before using it in your house. Check if the cables get warm (or even hot) - they should not. (Leave them plugged in for 2-3h and check every 20 minutes or so).
- If you want to connect the cables at the plug: buy a good plug, where you would be able to pin two wires into one connector. Probably one, which would have two channels for the cables, If not: drill a hole of appropriate size, so that your second cable will move in under high pressure. If it should be loose: apply hot glue or similar.
- If you want that one of the devices should be connected to the other: decide which goes first, drill a hole in it, then continue as above.
- If you want to connect the cables somewhere in the middle (which I do not recommend), empty the part of the cables where you want them to connect out. Prepare your insulation (shrink tubes, or fluid insulation), solder the cables and insulate.