Do I need to use "to" in sentences such as these?

Solution 1:

Your examples are wh-clefts. In the words of the ‘Longman Student Grammar of Spoken and Written English’ wh-clefts consist of:

a clause introduced by a wh-word, usually what; this clause has its own point of focus, usually at the end of the wh-clause

a form of the verb be

the specially focused nominal element

The specially focused element in the first example is to go to the cinema. This takes the form of a to-infinitive, because as a non-cleft it would occur as I want to go to the cinema, and not as I want go to the cinema.

The specially focused element in the second example is shut up and dance. This takes the form of a bare infinitive, because as a non-cleft it would occur as You should shut up and dance, and not as You should to shut up and dance. That in turn is because the bare infinitive is required after modal verbs such as should.

Solution 2:

Yes you absolutely need to in the first example. You cannot say:

What I want is go cinema.

This is not a case where to is functioning as a preposition that may be omitted. To go is the infinitive form of the verb, which is the form required by that sentence.

The second one should not have the word to at all. This sentence is an imperative (a command or a directive). You can distill that sentence down to:

Shut up and dance!

As for use in a formal paper, I wouldn't use either of them. I would say:

I want to go to the cinema.
Just shut up and dance.

I don't know what type of formal paper you're writing, but this sounds like dialogue to me.

Solution 3:

What I want is to go to the cinema. (grammatical)
What I want is go to the cinema. ( means ungrammatical)

As Barrie has pointed out, both of the example sentences are Wh-Cleft constructions.
That means they have been done things to. This sentence is from the prototype

• I want (for me) to go to the cinema. (Equi: same subject for want and go) ==>
• I want to go to the cinema. (Wh-Cleft) ==>
• What I want is to go the cinema.

The infinitive complementizer to in this sentence is still needed
to mark the infinitive clause to go the cinema
on the right side of the fulcrum of cleavage is
(the fulcrum be is inserted by the Wh-cleft rule);
this balances the embedded What-clause what I want
on the left side of the fulcrum.
It's that construction balanced on a fulcrum that marks a cleft sentence.

In the second case, however,

What you should do is shut up and dance.

the original has no infinitive complementizer to; on the contrary,
the infinitive forms do, shut (up), and dance
are governed by the modal auxiliary should, which is deleted in the downstairs clauses
by conjunction reduction (and not by Equi).
So there is no infinitive complementizer to.

• You should shut up and (you should) dance. (Conjunction Reduction) ==>
• You should shut up and dance. (Wh-Cleft: adds active pro-verb do) ==>
• What you should do is shut up and dance.