How do I say these things (more info below) in French?

I am going to + place = je vais à + place (remember that "à+le" is "au", that "à+les" is "aux" and for most feminine country and region names, "à" is replaced by "en")
Examples :
- I'm going to Rome = je vais à Rome
- I'm going to Italy = je vais en Italie
- I'm going to the museum = je vais au musée
- I'm going to the toilets = je vais aux toilettes
etc

I am going to + verb = je vais + verb in the infinitive
That's called the near future (futur proche). It's a grammatical construction that can be compared to a tense.
Examples :
- I'm going to fall = je vais tomber
- I'm going to need help = Je vais avoir besoin d'aide
etc

In order to + verb = afin de+verb in the infinitive/pour+verb in the infinitive
Examples :
- I'm sitting in order to relax = je m'assois pour me détendre
- I'm sitting in order to relax = je m'assois afin de me détendre

so + phrase = afin que+phrase in the subjuncitve/pour que+phrase in the subjunctive.
Examples :
- You're cheating so I can't win = Tu triches pour que je ne puisse pas gagner
- You're cheating so I can't win = Tu triches afin que je ne puisse pas gagner
Be careful, if the subject of the main and dependent clause are the same, using this formulation sounds bad, it's better to use the first two formulations (without "que") so the dependent clause doesn't have a new explicited subject.
Example :
- Je cours pour que je sois à l'heure (correct but sounds bad because the verbs "courir" and "être" both have a different "je" as a subejct)
- Je cours pour être à l'heure (correct and sounds way better because the verbs "courir" and "être" have the same "je" as a subject)

I want to give X a Y = I want to give a Y to X
So in French, that's "je veux donner un Y à X"
Therefore, "I want to give my brother a slap in the face" is "Je veux donner une gifle à mon frère" (I want to give a slap in the face to my brother).

(Disclaimer: Not a fluent speaker)

"I am going TO... (Dans? À?) ". Neither. This is called "futur proche" and it's formed with conjugating ALLER plus the infinitive. No additional preposition.
Example: Nager = to swim
Je vais nager = I am going to swim.
(Edit: I interpreted your question as "I am going to [verb]" but I see you might have meant "I am going to [location]".


"in order to / so"
I think in most cases you'd use a construction with "pour".
"I need glasses in order to read" = J'ai besoin des lunettes pour lire.

Your third question seems to be about the indirect object pronoun for "him". That would be "lui", and your word order was correct. So "Je voudrais lui donner" (which is more "I would like" than "I want" = Je veux)

Edit: I see while I was writing that, Tangi gave a more in depth answer. Tangi is a native French speaker.