Ich helfe mit, damit der Reparaturversuch weitergeht.

Yes. mithelfen usually means to help out, to join in helping, or to do your part in a shared effort.

So:

• Ich helfe. = I help / I am helping.
• Ich helfe mit. = I’m helping out / I’m pitching in.

In this sentence, mithelfen suggests that the speaker is contributing to a larger effort, not necessarily doing everything alone.

Because the verb is mithelfen, which is a separable verb.

In a normal main clause, German splits separable verbs:

• infinitive: mithelfen
• main clause: Ich helfe mit.

So helfe takes the usual verb position, and mit moves to the end of the clause.

Compare:

• Ich will mithelfen.
• Ich helfe mit.

Yes. helfen usually takes a dative object:

• Ich helfe dem Techniker. = I help the technician.

But the object can be left out if it is obvious from context or not important.

So Ich helfe mit is perfectly normal even without saying exactly whom you are helping.

Also, the mit here is not a separate preposition meaning with. It is part of the verb mithelfen.

Here damit means so that or in order that. It introduces a purpose clause.

So the structure is:

• Ich helfe mit = main action
• damit der Reparaturversuch weitergeht = purpose/result intended by that action

In other words: the speaker helps out so that the repair attempt can continue.

Because um ... zu is usually used when the subject of both actions is the same.

For example:

• Ich helfe mit, um das Problem zu lösen.
  = I’m helping out in order to solve the problem.

Here, the person doing the helping is also the one solving the problem.

But in your sentence, the subject of the second clause is der Reparaturversuch, not ich:

• Ich helfe mit
• damit der Reparaturversuch weitergeht

Since the subject changes, damit is the natural choice.

Because damit introduces a subordinate clause, and German normally separates subordinate clauses with a comma.

So:

• Ich helfe mit, damit ...

That comma is required in standard German.

Because the verb is weitergehen, another separable verb.

Its basic forms look like this:

• infinitive: weitergehen
• main clause: Der Reparaturversuch geht weiter.
• subordinate clause: ..., damit der Reparaturversuch weitergeht.

In a subordinate clause, the conjugated verb goes to the end, and with separable verbs the prefix usually stays attached there. That is why you get weitergeht as one word.

Because damit creates a subordinate clause, and in German subordinate clauses the finite verb goes to the end.

So:

• main clause: Der Reparaturversuch geht weiter.
• subordinate clause: ..., damit der Reparaturversuch weitergeht.

That final verb position is one of the most important word-order patterns in German.

Reparaturversuch is a compound noun:

• Reparatur = repair
• Versuch = attempt

So literally it is repair attempt.

The gender comes from the last part of the compound, which is Versuch. Since der Versuch is masculine, the whole compound is masculine too:

• der Reparaturversuch

And it is capitalized because all German nouns are capitalized.

It is the same written word, but not the same function.

German damit can mean:

1. with that / with it

   • Ich schreibe damit. = I write with it.

2. so that / in order that

   • Ich helfe mit, damit der Reparaturversuch weitergeht.

In your sentence, it is clearly the second one, because it introduces a full clause.

Yes, that is grammatically possible. But it sounds a bit less specific.

• Ich helfe = I help
• Ich helfe mit = I help out / I join in helping

Using mithelfen emphasizes that the speaker is contributing to a joint effort. In this sentence, that nuance fits very well.

Yes. weitergehen often means to continue, to go on, or to carry on.

So:

• Der Reparaturversuch geht weiter. = The repair attempt continues / goes on.

It is a very natural German choice here. German often prefers verbs like weitergehen where English might simply say continue.