I have never done that it may be possible using the Advance installer options. I use the advance options to keep older version of Photoshop installed. I do not remember what other options there are in there I would think it is possible to change where it install Photoshop it would be in there. .
Sure, that should be no problem. Whether it will speed up loading? Not certain - depends on the USB speed. It will probably load up slower. It's a better idea to free up space by removing data files - source files of work that you've done, and other larger data files.
Generally it is not a great idea to install your major software on an external drive.
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Most of it still ends up on your system drive, under your user account.
And that's where the junk accumulates, from all your applications. A thorough sweep of your user account can often free up 10 - 60 gigabytes or more. Empty out all app caches (or move them). Bridge alone can be responsible for 20 GB.
In addition, run disk cleanup to remove all kinds of other leftover system files.