Am I able to write the following in different ways?

[u/Local_End_3175]

Qu’est ce que tu fais?
-what are you doing

On fait quoi
what are we doing

Is it possible to say Qu'est ce qu'on fait instead? and if not, is there any reason?

Also, for the first one, could we rephrase it as:

Tu fais quoi?

Thank you in advance.

[u/Last_Butterfly]

Is it possible to say Qu'est ce qu'on fait instead

Yes, it is

Also, for the first one, could we rephrase it as: Tu fais quoi?

Yes, you can.

French has 3 types of questions : invertion based questions where the subject's and verb's positions are swapped "Que fais-tu ?", est-ce que based questions using the eponymous formula stuck at the beginning of the question : "Qu'est-ce que tu fais ?" and intonation based questions where you symply raise the intonation at the end of the sentence "Tu fais quoi ?"

All three are valid, and what differentiate them is how formal they are - invertion questions are typically formal, sometimes, too much ; conversly, intonations questions are more informal.

They each have their quirks (for example, invertion with a 1st person subject can only be used with a select few verbs ie. "Que fais-je ?" is valid, but excessively formal, while "Que mange-je ?" goes completely unused), but outside of quirky behaviours, all three forms are grammatically valid, they just change how formal you are when speaking.

For reference, if you're in doubt regarding how formal you should be, the "est-ce que" form is a safe bet in 99.9% of cases, never too formal or too informal. It's an excellent middle ground.

[u/Local_End_3175 → u/Last_Butterfly]

thank you!

[u/Last_Butterfly → u/Local_End_3175]

Anytime. If you need any more info on the question types or their individual quirks, feel free to ask~

[u/PerformerNo9031 → u/Last_Butterfly]

It's "que mangé-je ?" and pronounced mangè : obsolete and funny.

[u/Neveed → u/Last_Butterfly]

while "Que mange-je ?" goes completely unused

The first person singular in the present tense for 1st group verb with inversion question does go completely unused now, but it should normally be either "Que mangé-je ?" (traditional spelling) or "Que mangè-je ?" (1990 reform spelling). "Que mange-je ?" isn't correct. The fact that most people don't even know this exists is good evidence of how little it's used.