Definitely worth getting the battery checked. It can be done for free at Halfords or similar. If it's not the battery then a diagnostic check is recommended.
I'd definitely put it down to outside temperature. I know my older A6 takes longer to start in cold mornings. You could always get a diagnostic check to see if everything is working as it should.
Welcome to the forum you'll find the members on here are a friendly and helpful bunch 🙂
The V6 diesel A4 would have had the same engine as my A6.
Nice to see youve had a few different Audi's 🙂
This link might help:
https://www.audi-sport.net/xf/threads/dab-radio-signal-suddenly-very-poor.349691/
It could be the amplifier. Or have you fitted a dash cam lately?
Honestly I'm not sure, but I imagine both should work if you've got 2 fitted. If it's not working it could be down to a failed sensor or broken wire maybe?
They'll bolt in but you'll need the wiring harnesses and then coded to the car. The problem is it might cost more in the first place then change the bulbs or ballasts in the first place.
You could always try and source second hand xenon headlights.