This has to be possible I just don't know how.
Don't be so sure. The only emulator I'm aware of which is widely available for any model of Pi is QEMU, and while that definitely can be used to emulate the BCM 2835 based models (A/B/+/0), and there is some indication that it might be usable to emulate the BCM 2836 (i.e., the Pi 2), the status of the BCM 2837 (i.e., the Pi 3) is unclear. By this I mean I don't think there is an image specifically for it, but should be plausible to use a generic 64-bit ARMv8 (aka. aarch64) image and run Raspbian within that.
[You don't actually mention an OS, but hopefully you can extrapolote -- I'll just refer to Raspbian.]
This would mean not using the real Pi kernel and not having access to certain physical features which would be a bit meaningless in an emulator anyway, except perhaps to test software exploiting those physical features (the GPIO, the camera interface, the onboard wifi and bluetooth).
Whether this would seriously impact what you want to do I'm not sure. Note that some pi-specific configuration involves files on the boot partition, so you will have to make arrangements for that if you want to use, e.g.,
raspi-config, which by default is run the first time you boot Raspbian.
It's worth noting that since Raspbian (and as far as I am aware, all the other potential operating systems; Windows IoT may be an exception) is identical regardless of model except for the kernel, you could use one of the other QEMU images and the only thing it will make any difference to that I can think of off-hand is configuring the serial console, which uses a different device node on the 3.
I am fairly certain you should be able to download, install, and configure most software using the distro package manager (on Raspbian,
apt) that way, but getting this all working correctly, especially without any previous experience using and configuring a Pi, is very unlikely to be easy.
Doing this will keep me from dragging my mouse and keyboard and internet connection back and forth to the other room. Any thoughts?
Yes: If you follow this path you will be begging and pleading to "drag" a keyboard from room to room instead within a few hours.
Put another way, it may not be impossible, but it is going to require much more effort, not less.