Da stecken sie sich dann halt auch an

Da stecken sie sich dann halt auch an. It has nothing to do with stecken. The way german verbs like, for example, anstecken, anziehen, ausfahren, hereinkommen, hinausgehen, herablassen, durchfahren are used when conjugated, like this:

• Ich stecke mich niemals an.
• Du steckst dich bei den anderen Menschen an.
• Heute steckt er sich an.
• Wird sie sich anstecken?

The halt in that sentence is just a stupid fillword, which usually means that the outcome is obvious or likely to be expected, given the context. In this context, it means that for the person speaking it's obvious that they are gonna be infected because they are kids, playing around, not being as careful.

Halt is Umgangssprache and is completely fine to be used, is often used, but is not a formal way to say things and some people tend to overuse it. And in my opinion, it has a somewhat negative tendency.

• Wer nicht hören will der muss (halt) fühlen.
  → One that doesnt listen will have to feel it.

• Mach halt die Tür zu.
  → Close the door, (I told you already).

• Dann' mach halt.
  → Do it if you have to.

• Das kannst du halt nicht machen.
  → You can't do that, obviously.

• Does "da sich stecken" mean "to get oneself into there", "to go there" ?
• What is the subject of "anhalten" in the second clause? Does this verb mean "to stop", "to bring to a halt" or something else in this context?

None of the above.

You have a confusion about how those words relate: stecken and an come from (sich) anstecken and not from (sich) stecken or anhalten.

Anstecken is torn apart in that sentence structure, it could have been also (probably clearer for a non native speaker):

Dort ist es, wo sie sich halt auch anstecken.

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• Is "auch" a modal particle in this context ?

No, but halt is. auch refers to the location (die Schule).

The correct translation might be

That's where they (the kids) finally get infected also (at the school).