Fully vaxxed is two jabs. If he has those he doesn't need a supplemental test to get in.
The law permitting restrictions is due to expire on 20 March, and it's unclear right now what will happen after then.
It's useful to know about 2G, 2G+ and 3G when you're here. 2G means only people who have 2 vax or recovered (with a recovery document) within the last 3 months.
2G plus means only vaxxed or recovered plus another requirement - usually that you have the booster or a recent LFT
3G means vaxxed (2 doses), recovered or tested.
Each federal state has different requirements, and each category of venue within those states.
Every indoor public space requires medical masks, most require FFP2. I can never remember which is which so we all wear FFP2 for everything, as do about 70% of people (the others wear disposables when they are allowed). Public transport in eg Berlin is FFP2 and 3G, so he would need to get the correct mask at the airport or before. Shops are probably going from 2G with any medical mask to 3G with FFP2 in the next few days.
But so much can change in the next month, the rules are unlikely to get more strict unless a new variant emerges though.
PS: Berlin is full of pop-up testing stations, and residents can get free LFTs there (or buy them from shops for about 2 euros each) - tourists have to pay for an official LFT though. If your DS needs a test to get to a 2G + venue he'll have to pay, as self tests are not valid proof for entry.