Is this a grammar error? ( Russian )

I am currently at lesson 20 of Who Is she. I came across what looks like a grammatical oddity/inconsistency. I could be wrong, so I need a little help here.

давайте действовать быстро (the 1st verb is in the imperative and the 2nd one is in the infinitive, I think the sentence says " {you} let us go ! " )

давайте пустим Свету в квартиру сейчас же. ( here the 2nd verb is conjugated in the first person plural )

There is another instance in an earlier lesson where the verb appears to be in the third person singular while it should have been in the 2nd person singular indicative, but I will wait for an answer for this one first.

давайте, давай - let’s (let us)

давайте - is used either formally as polite form (both plural and singular (i.e. when speaking to one person)) or informally in plural (let us guys…)

давай - is used informally when speaking to one person (let us both) or giving orders to someone (to one person, may not sound very polite).

давайте действовать быстро - let’s (all of us) act fast; let’s (both of us, polite form) act fast
давай действовать быстро - let’s (both of us) act fast (informally)
давай действуй быстро - [you], act fast! (informal)

In first two cases 2nd verb should be in plural (there are 2 or more persons who will act).