Can I say
Let we do it
Instead of
Let's do it
One more question. Because let's is a contraction of let us, so can 'us' be a subject? Isn't 'us' only used as object?
Can I say
Let we do it
Instead of
Let's do it
One more question. Because let's is a contraction of let us, so can 'us' be a subject? Isn't 'us' only used as object?
Nope, and nope.
Subject pronouns (I, we, you, he, she, it, they) come before the verb in sentences and clauses. (Exception is gerunds and infinitives, this is not one of those exceptions).
After the verb, if a new clause has not started for any reason, you need to use object pronouns (Me, us, you, him, her, it, them).
English imperative or "command" mood usually omits the you in front of the verb, which is why it's not there in "Let us X..." It can still be included for clarity.
If gerunds or infinitives have subjects, then object pronouns are used (because technically they can't have subjects, I think). If the verb doesn't end in -ing then you aren't using a gerund, and if to isn't in front of the verb, then you aren't using an infinitive.
I walked to the park.
Me walking to the park was a bad idea.
We walked to the park.
Us walking to the park was a bad idea.
To go to school was considered the smart thing to do.
Me to go to school at 2:00 is what she was planning. (I think this is correct but sounds really awkward, the gerund preferred here.)