What level find familiar is cast at depends on how the warlock casts it.

When she gains the Pact of the Chain class feature, the warlock learns find familiar. It doesn't count against her spells known, but nothing in the ability text prevents her from casting it with her spell slots. If she casts it with her spell slots, it works exactly like any other spell she knows how to cast. If she is 5th level when she casts it, she casts it as a third-level spell.

In addition, the warlock gains the ability to cast find familiar -- and only find familiar -- as a ritual. When a spell is cast as a ritual, it always uses the normal spell level of a spell. When the warlock casts find familiar as a ritual, she casts it as a first-level spell no matter what her warlock level may be.

There likely isn't a RAW answer, but...

Being permanently mute is not a condition that the core rules represent, so any answer here is going to be speculative and highly subject to the DM's rule on the matter.

However, I would encourage a DM ruling on this to allow it, particularly because the character is taking on a significant disadvantage that isn't compensated for anywhere else in the rules. So giving a free "out" isn't exactly letting the character get away with anything.

A more important question for you before you decide to volunteer your character for this disability is how you plan to use spells that have a verbal component. A character who can't speak (due to a gag or silence spell) generally can't use verbal spells, and speaking through your familiar may or may not make up for this limitation, according to your DM.