Does anyone know what happens if say, you apply for a couple of form 1s. Then in another three, four weeks or so, you send in applications for a couple more forms.

If the first two forms go into "FBI check" status, will all subsequent forms that follow do the same until the FBI check is cleared? Are they treated differently/independently of each other? Does the same examiner get all of them?

Is this how batch approvals work?

No, they are treated separately/independently.  Forms have routinely cleared/been approved while earlier forms were still held up.

The forms are currently assigned to different examiners after the FBI check has cleared based on workloads.

Doesnt matter, just wait. Nothing you can do to speed anything up..... My Form 1 took 8 months, M16 on form 4 took 18m mo. Gemtech HALO took 7 months, CAC 9 took 4 months, M11A1 took 13 months.

Someone elses experience makes no difference to yours or indicates how things are approved. Depends on the stack of forms the have, not what someone says here.

There is if you're a LEO and the item being transferred is for official use (though not necessarily exclusively).
If this applies to you, IM me for further info.