Rechargeable AA batteries will be 1.35v when full, 1.2v nominal.
Fresh non-rechargeable batteries will be 1.65v, 1.5v nominal.
NiMH cells don't like being charged when they're not fully discharged - if you want them to last longer it's a good practice to fully discharge them and then charge them. No, 0.5A is a bit much. I would say 100-200mA would be safe charging current for modern AA cells.
You have the service manual here :
https://archive.org/details/manual_D6350_SM_PHILIPS_EN/page/478/mode/2up You can see on 2nd page (pages 478-479) in the bottom corner that the transformer output is rectified with diodes and passed to some capacitors and when you put the AC cable in socket, a switch is pressed disconnecting the batteries.
So as far as the cassette player knows, it doesn't differentiate between ac input and batteries, so there's no point going with 8 cells to get 12v DC, you won't get better performance or anything.
You can also see that they use a time delay 630mA fuse on the secondary of the transformer so the current is probably less than 4-500mA , making it 12v x 0.5A = 6 watts maybe, max.
I wouldn't use NiMH, I would use lithium cells. Two lithium cells in series would give you 8.4v peak voltage, 2 x 3.7v = 7.4v nominal. 6 rechargeable C batteries x 1.2v = 7.2v ... if the player works now with 6 rechargeable cells, it will work with 2 lithium cells in series.
So that leaves you with finding a small charger board / module that can charge two lithium cells in series. Here's one :
https://www.ebay.com/itm/364722753475This particular charger board is powered with 5v from USB. There's other models that need a voltage slightly higher than the voltage needed to charge the 2 cells (for example 9v, because the peak voltage for 2 cells in series is 8.4v)
Here's a link for chargers that accept up to around 22v and can charge 1, 2 or 3 in series (select from the dropdown list that you want the module for 2 cells in series ) :
https://www.ebay.com/itm/134024292844If you want , you can add a step up regulator to boost 6v ... 8.4v to 9-12v, but that step-up regulator may introduce some noise in the audio, from the high frequency switching.
Here's an example .. it takes any voltage up to 24v and outputs 12v :
https://www.ebay.com/itm/274714390574