If I wanted to suggest that we speak French, in looking for a common language, could I simply say "Parlons français?" and use intonation to make it a question, or would that be grammatically incorrect?
What would be a polite way of requesting it, like a translation of "Shall we speak French then?" Google suggests "parlerons-nous français alors?" ("will we speak French then" as its back-translation) which looks right to me, although I wouldn't have gotten that verb form and inversion guessing. (It also offers "allons-nous parler français alors?" ("are we going to speak French then") ; any thoughts on whether this would be more or less formal? I feel like the first one is better for my purpose ; not sure if either is quite spot on. I'm feeling like it should be like a "parlions" or "parlerions" in there but not really sure why.)
Because this uses the imperative. So basically the speaker is giving the order to speak French, but in the form which includes the speaker himself/herself. Since there is no imperative form in English that includes the speaker, "Let's speak French" is the closest translation possible here.
Just another learner with a thought:
Unlike with Spanish (and other Romance languages?) you cannot usually drop the subject pronoun in French. "We speak French" = "Nous parlons français."
The one exception seems to be in the imperative. Here the pronoun is dropped "Parlons français". That is how you know it is imperative!
However, if it is a verb that takes a reflexive pronoun:
With a positive imperative the stressed pronoun comes after the verb with a hyphen, e.g. "tais-toi";
with a negative imperative, the usual pronoun is inserted between "ne" and the verb, e.g. "ne te blesse pas."
Caveat: For all this about the imperative, you would be better off to read georgeoftruth and Sitesurf's notes on this skill.