By default it will power on when connected so you could use a simple switched live... of course you may have to be careful not to turn off the ignition whilst the DB410C is writing to its eMMC (or design your software so that the critical partitions can be mounted read-only).
There might be scope for more exotic approach but this will require carefully timed management of the power buttons (power button can distringuish short press, 3s press and 10s press).
FM_RX_ANT is routed to pin 15 of the analog audio connector... other than that I'm afraid I have no clue (I also noted that the hardware manual says that FM radio "an optional addition that has not been tested and is not officially supported"). I'm afraid you might just have to experiment here.
You could also try interfacing one of the RTL-SDR USB dongles since that will allow you a greater range of signals and some future proofing for jurisdictions where FM will be turned off soon. RTL-SDR is a family of cheap TV receiver devices that can also receive DAB and FM signals (I'm afraid getting these running is out-of-scope for this forum but the RTL-SDR community might be able to help).
Board should not be drawing more than 24W. In fact IIRC 17W of this budget is for powering downstream things either via USB or via the LS connector. In other words you can use quite a small fuse; 2A is a sane default but you could probably tighten things further if you want to.
I don't have any particular recommendation but you will need an external antenna, the PCB antena will not be sensitive enough.
The voltage input for this board ranges from 6.5v to 18v so there should be no point when the car voltage is running out of spec (car looms can are 12v nominally but can reach up to about 15v, is that right?).
I'm afraid I don't know the power draw figures that would let you calculate how long you can run the board for on a battery.