Unsere Babysitterin kommt heute Abend, damit wir ins Kino gehen können.

Damit introduces a purpose clause: it explains the purpose of the babysitter coming.

German usually uses damit when the subject of the main clause and the subject of the purpose clause are different.

• Main clause subject: Unsere Babysitterin
• Purpose clause subject: wir

So damit is the right choice here.

By contrast, um ... zu is normally used when the subject stays the same, for example:

• Wir sparen Geld, um ins Kino zu gehen.

There, wir is the subject of both parts.

Because damit starts a subordinate clause, and in German subordinate clauses are normally separated from the main clause by a comma.

So:

• Unsere Babysitterin kommt heute Abend, damit wir ins Kino gehen können.

That comma is required.

After damit, German uses subordinate clause word order, which means the conjugated verb goes to the end of the clause.

Here, the conjugated verb is können.

So the structure is:

• damit
  • subject + other elements + verb(s) at the end

That gives:

• damit wir ins Kino gehen können

Not:

• damit wir können ins Kino gehen

In a clause with a modal verb, the main infinitive (gehen) comes before the conjugated modal (können) at the end.

Because können is a modal verb and gehen is the main action.

• gehen = to go
• können = to be able to / can

So ins Kino gehen können means to be able to go to the cinema.

This makes good sense here: the babysitter’s arrival makes it possible for us to go out.

Yes, that is grammatically possible.

• ..., damit wir ins Kino gehen.
• ..., damit wir ins Kino gehen können.

The version with können emphasizes that the babysitter makes it possible for us to go. That is very natural in this context.

The version without können sounds more like so that we go to the cinema, focusing on the intended result rather than the ability/opportunity.

German often uses the present tense to talk about the future when the time is already clear.

Here, heute Abend already tells you that the action is in the future, so kommt is completely normal.

• Unsere Babysitterin kommt heute Abend.

Using wird kommen is possible, but it is usually less neutral and often sounds more formal or more emphatic.

Literally, it means today evening, but natural English would usually say this evening or tonight.

So heute Abend can often match either of those, depending on context.

Ins is the contraction of in das.

• in das Kino → ins Kino

With a verb of movement like gehen, German uses in + accusative to show movement into a place.

Also, ins Kino gehen is the normal German expression for to go to the cinema / to go to the movies.

So ins Kino is both grammatically correct and idiomatic.

Babysitterin is the feminine form, meaning a female babysitter.

The ending -in is a common way to form feminine nouns for people:

• Lehrer → Lehrerin
• Freund → Freundin

Because Babysitterin is feminine and is the subject here, the possessive form is unsere.

If it were a male babysitter, you would normally say:

• Unser Babysitter kommt heute Abend ...

Because in German, all nouns are capitalized.

So:

• Babysitterin = noun
• Abend = noun
• Kino = noun

This is a standard spelling rule in German, not a sign of special emphasis.