No. There is no other way to get an authentication token other than authenticate
Perhaps you could simply encrypt the admin password in your config file and decrypt same in your code before using it? There are any number of design patterns (that don't have much to do with Tableau) that deal with "keeping a password safe". You can read about them, pick your favorite, and apply the technique in question.
Correct me if I am wrong, but X-Tableau-Auth equals to workgroup_session_id - thus you can obtain it via trusted auth.
I did not test it but seems logical for me
Totally correct. Didn't mention that as I figured most won't want to go after the value inside the cookie...and the fact that knowing how to store / encrypt / protect sensitive information like a password should be part of every developer's toolbox - so it doesn't make a lot of sense (to me anyway) to do something "un-natural" just to avoid storing the password. Just my opinion, though.
Yes, you are right, but still, in some circumstances it is easier and more secure to use trusted authentication than a password based one. Even if you store the passwords in the config files you have to have the decrypt algorithm on the same box (to be able to use your tool), thus, if you have access to the crypted config, you will have be able to see the source of the program which decrypts it -> you have everything to crack it.
Also, not sending plain text passwords on the network (if ssl is not enabled on the server) is generally a good idea