What does “can’t mix neutrals” mean, and how would neutrals get mixed

In the main panel, ground and neutral are connected, so you're free to attach to whichever bar is most convenient. There are two bars because if it was used as a subpanel ground and neutral would have to be kept separated.

The main breaker will cut power to the bus bars and breakers, but the lugs and feeder cables near the top of the panel will still be hot. Shouldn't be too hard to avoid those since the top slots are most likely full anyway and you'll be coming up from the bottom, just pay attention.

I'm not sure what code requires for routing, but it's probably best to drop the wire down in the neighboring cavity and cross through the stud at least 6-12" below the panel before coming back up to keep things nice and tidy. Or for a surface-mounted panel just route the cable neatly and anchor with cable staples:

This behavior is because arc-fault breakers have an interesting quirk: they also provide a small degree of ground fault protection as well! (They are not GFCI's because their ground-fault trip threshold is set to the 30mA used for equipment protection (GFPE) devices as opposed to the 5mA ground-fault (differential) trip of a Class A personnel protection GFCI.)

You will need to contact your electrician and tell them to get their neutrals straight. Tell him he violated NEC sections 200.4:

Multiple Circuits. Where more than one neutral conductor associated with different circuits is in an enclosure, grounded circuit conductors of each circuit shall be identified or grouped to correspond with the ungrounded circuit conductor(s) by wire markers, cable ties, or similar means in at least one location within the enclosure.

300.3(B):

Conductors of the Same Circuit. All conductors of the same circuit and, where used, the grounded conductor and all equipment grounding conductors and bonding conductors shall be contained within the same raceway, auxiliary gutter, cable tray, cablebus assembly, trench, cable, or cord, unless otherwise permitted in accordance with 300.3(B)(1) through (B)(4).

and 310.10(H)(1):

General. Aluminum, copper-clad aluminum, or copper conductors, for each phase, polarity, neutral, or grounded circuit shall be permitted to be connected in parallel (electrically joined at both ends) only in sizes 1/0 AWG and larger where installed in accordance with 310.10(H)(2) through (H)(6).