Personal advice, go for it - there's no harm in applying and interviewing at other positions (I've never quit a job before looking for another, I've always had another job lined up first), if you get an offer you then have a decision to make - when you hand your notice in, depending on the company etc. they may make a counter-offer to keep you - don't depend on them making it (ie make sure you have a concrete offer/signed offer letter before handing your notice in) but you need to decide if there's any counter-offer they can make that will make you happy to stay.
Personally, I've never taken a counter-offer (there's a reason I'm leaving, and an offer made when I say I want to leave isn't going to work out long-term) but if they offer to bring your wage in-line with your colleagues make sure it's in writing, along with anything else you want (ie additional training courses/certifications with a time-frame for that to be actioned in)
And no, your boss can't fire you for looking elsewhere - depending on how comfortable with your boss you are, it might be worth bringing it up before you look ("hey, just to let you know I'm unhappy about x, y and z, is there anything the company can do for me, otherwise I'm unfortunately going to have to look at other options available to me")