10kBTU = 2.93kWh.
Heat pumps typically provide 2-4x as much heating/cooling as they consume in electrical power, so if we take a 3x factor, then a 2.93/3= 977W draw seems right in the ballpark. 8x110 = 880W also gets you into the same area. 1000W is a safe call.
Start-up current for ordinary ACs is highly variable, but commonly 2-6x continuous current. Unless you have more details, and if we take the average of 4x, then the initial current draw of your AC could be 4x1000= 4kW which is double the capacity of your inverter. This will pose problems if your batteries are low, the sun is shining, and you turn on the AC — the inverter may trip.
Anyone going off-grid should buy a variable speed "inverter" AC unit. They ramp up to speed slowly so aren't as punishing to off-grid electrical systems as the ordinary — fixed speed — types that expect 100-200A to always be available from the mains.
6x 100Ah batteries = 600Ah.
600Ah @ 12V = 7200Wh = 7.2kWh.
You haven't specified the type of battery, so useful capacity is unknown. Let's assume your batteries are AGM and you want them to last at least 5 years, so you don't discharge them below 50%. So DoD is 50%.
3.6kWh of usable power will run your lights, laptop and AC for ~3 hours once the sun starts to sink late in the afternoon. That's not exactly a long time. Hope you plan to go to bed early.
6x 325W panels = 1950W.
1950W @ 12V = 162.5A.
162 >>> 40 so, obviously, you're not going to have a 12V battery bank.
1950W @ 24V = 81.25A.
81 >> 40 so, obviously, a 24V bank won't work either.
1950W @ 48V = 40.625A.
Now we're in the ballpark... but you can't create a 48V battery bank using 6 batteries — you need a multiple of 4.
Q. 1) I'm unsure if I should run the PV panels in 3 parallel series (3 groups of 2 @ 48V 15A) or just run them parallel at 24V 30A.
A 325W panel that outputs at 24V should be producing 325/24= 13.5A.
3 in parallel x 2 in series should generate 40.5A@48V.
6 in parallel should generate 81A@24V.
Your current values seem to be out by a factor of 2.7 — you might want to work out why.
Q. 2) Likewise similar question for the batteries: should run them in 3 parallel series (3 groups of 2 @ 24V 15A), can I just run them parallel at 12V 30A, or if I'm running the PV array at 48V do I need to match the 48V so that the battery controller is 48V both in/out?
Your bottleneck is the 40A charge controller. If that's the maximum current it can feed into the batteries, and you want to be able to absorb all 1950W of power from the panels, then you need to be outputting at no less than 48V to the batteries.
Given 12V batteries, you need 4 of them in series to get to 48V, so your battery bank will consist of one or more such strings. That means either 4, 8, 12, 16, 20, or n*4 batteries.
You can't turn 6 batteries into a viable 48V bank, so get two more — 8 batteries in total — and have 2 parallel strings of 4 batteries in series... at least.
No, the incoming voltage from the PV array does not need to match the outgoing voltage to the battery bank. In this case the charge controller is so marginal that you don't have any other option. If you went with a 60-80A charge controller (or two 40A charge controllers) you would have more flexibility, would get better charging efficiency, and would be able to take advantage of cloud-edge effects when the current can briefly spike up to 50% above normal.