Can I use an iPhone 5s without a SIM card?

I'm planning to buy an iPhone 5s but I need some help first.

1. Can I set it up and use it to play games, and watch YouTube without a SIM card?

2. If I transfer my SIM card from my old iPhone 5s to the new one will the data appear on my phone bill and say something along the lines of "new iPhone connected" or something?

You can use it just on Wi-Fi. But you have to have Wi-Fi to use it. If you're off Wi-Fi, you can't connect, download, stream, etc.

When you go through the initial setup process of the phone, it will require a SIM card. There's no way around it. The SIM card has to be from the carrier that the phone is locked to, OR if the phone is unlocked, any compatible SIM.

However, you only need the SIM card once for that initial setup. Once it is setup, you can remove it and the phone will still work (obviously except for the phone part and mobile data and texting). Though, it may occasionally nag you that there's no SIM inserted.

As for your second question, I don't think it will.