As Fernando mentions, the answer to the question in your 1st sentence is no, because some such complexes have unsolvable word problem in their fundamental group presentations.

The answer to your question about replacing homotopies by linear homotopies is yes -- just think through what the Seifert VanKampen theorem says about your 2-complex. The problem is you don't know what linear homotopies to apply as there's too many of them. One of the "killer" linear homotopies is one where you introduce cancelling pairs $gg^{-1}$. You don't know how much longer you need to make a word before you "spot" a relation. That kind of thing.

As Fernando mentions, the desire to have an algorithm to determine if two maps are homotopic, generally-speaking is impossible to satisfy. This is because some such complexes have unsolvable word problem in their fundamental group presentations. There are (many) complexes that do not have this kind of trouble, but what this means is that any algorithm you write will not work for general complexes.

The answer to your question about replacing homotopies by linear homotopies is yes -- just think through what the Seifert VanKampen theorem says about your 2-complex. The problem is you don't know what linear homotopies to apply as there's too many of them. One of the "killer" linear homotopies is one where you introduce cancelling pairs $gg^{-1}$. You don't know how much longer you need to make a word before you "spot" a relation. That kind of thing.