Shall we get started?
Is "started" an adjective here? I guess since in future tense we don't use past participle.
My opinion differs - I would consider it one of those participial adjectives.
As far as I can think of, the only way to use the past participle with a modal verb is with the present perfect - in other words, you need to include 'have'. Somebody may be able to correct me otherwise.
That would give you 'Shall we have started', which is grammatical but a bit unnatural.
If you swap in some other non-participial adjectives, they still work.
Shall we get naked?
Shall we get busy?
Shall we get happy?
(side note - here's a
link to Hugh Laurie doing an excellent version of "Get Happy". Alternately, here's an
audio-only version, without the weird choreography from the scene.)