A person with out of status can obtain temporary legal by applying for TPS. Yes, A person can travel if he/she obtains advance parole.
a, person can fuck dog, cat, mouse whatever with TPS but again its temporary(18 mths), After 18 mths it will keep on getting renewed( history says so).
A person can apply for TPS while he/she may have other nonimmigrant visa status( for them TPS will be extra layer of protection and also EAD if the person needs one)