@eventoowish No, you can't download only the EPs on a flash drive. They'll be installed wherever the base game is. You can however install the entire game on a flash drive using one computer, then plug that drive into a different computer. You'd still need to tell the EA App to "install" Sims 3 in that location on the new system, but the App will (probably) simply verify the files and complete the process.
For the issue installing to D, make sure you're installing into a folder on D, for example D:\Games rather than the root level of the drive. If this doesn't help, would it be possible to format the drive? It should be formatted as NTFS anyway, but if it already is, you could format it as something else and then back to NTFS.
Whenever you format a drive, its content is erased, so don't do this if the drive holds other data you don't want to lose. The alternative is to partition the drive and use the new partition. Sims 3 takes up about 32 GB with all packs installed, program files only, so a 40 GB partition would be fine for the game itself, and you'd need more space if you want to place your user files there too.
For the Alder Lake issue, try the Task Manager method instead.