I have found (often to my surprise) that much FOM (Foundations of Mathematics) is done assuming full blown ZFC set theory. You can certainly think of formal language theory, computability theory, model theory, etc... as subjects in ZFC just like any other subject such as abstract algebra or topology/geometry. There is no danger here as long as the context in which the results were derived is clear.

Meta-mathematical work is often done in ZFC first and then under weaker systems later.

For the foundations of formal language theory, the following ideas show up:

1. Universal algebra (here for 'free' monoids and basic logic involving simple equation only proofs, just variables and no connectives or quantifiers)

2. Term rewriting systems (take a look at the wikipedia article)

As for foundations, I have found that much FOM (Foundations of Mathematics) is done assuming full blown ZFC set theory. You can certainly think of formal language theory, computability theory, model theory, etc... as subjects in ZFC just like any other subject such as abstract algebra or topology/geometry. There is no danger here as long as the context in which the results were derived is clear.

Meta-mathematical work is often done in ZFC first and then under weaker systems later.