Correct! The term "illegal immigrant" presupposes that we already know why they're here (namely: to immigrate, i.e., to settle down). But some of them could just as easily be here for other purposes (earn some money for a season, then go home; sell drugs, then go home; plant bombs, then go home; etc.).
The most-neutral and thus most-accurate term is "illegal alien."
Regards,
Yes they are unworthy of being called immigrants.
Trespassers.
No; it is NOT!
is redundant!