I don't think it can send files to the computer, but I am not 100% sure. You would need to ask Lionel Debroux or another 68K expert. From what I remember, the only emulator that could send files to the computer was WabbitEmu, and sadly WabbitEmu doesn't have 68K calculator model support.
It can't record gif either. You need an extra program called CalcCapture and play with the settings.
To send a file from TIEmu to the computer:
2nd+Var-Link
F4 the files you want to send
Use F3 2 (Send...)
Go to the directory in which you want to save the file.
The resulting file will be a single file if you sent one, and a group if you sent more.
Edit: Darn, already answered.