Let's (ましょう)

ましょうか often isn’t proposing something for both of you, it’s one person offering to do something alone and checking if that’s wanted, closer to “should I…” than “let’s.” Context, especially who’s actually doing the verb, tells the two uses apart.

ろう/おう/よう is the casual version of ましょう, built by shifting the verb’s dictionary-form ending the same way the negative and past tenses shift, not by attaching ましょう and then stripping the polite part off. 話す becomes 話そう, 食べる becomes 食べよう, 来る becomes 来よう, each verb group has its own casual volitional shape to learn.