for a kid who will not quit

As a fellow dad of a 10 year old, appreciate that they haven't started having issues yet with the competitive side. I'd recommend watching some youtube videos to get a feel for the gameplay in multiplayer and/or get your feet wet in an evening after they go to bed. Then you at least have the basics down to have your son teach you how to play over discord/voice chat. I'd recommend picking one game at a time for BF as they are similar but have enough nuances that if you aren't familiar with them it could be difficult to bounce between them. 

I'm not a huge fan of 2042, but the solo/co-op vs bots will give you some familiarity with the maps and weapons (and allow you to slowly rank up, unlock weapons and their attachments) and get used to the gameplay all without the stress of multiplayer/competitive games. 

Thank you for the kind advices.
I face 2 constant issue - anticheat most of the times blocks one of us if we join in same time - I presume because of the IP's as we share same connection/network.

Second issue: is there a way to get a room/mission to rehears or at least how to make sure I get in same playgroud and the kid - I mean except choosing servers - once eliminated there is that randomness that kicks me into different areas or re-spawn with random players etc Do you know of servers where I can do this easier?

In BF2042 you should be able to join a "party" together which will barring very few circumstances keep you together. If you are using Portal I am unsure of if it will possibly split you, but regular/main menu All-out-Warfare Conquest/Breakthrough queues should keep you together. The Solo/Co-Op mode would also keep you together in the party to fight against bots in either conquest or breakthrough as well. 

I'm not aware of anti-cheat preventing two people on the same network from being able to play together, otherwise, I imagine these message boards would be lit up with complaints about it (and I haven't seen it mentioned, at least not in the BF2042 forum that I frequented almost daily since launch up until somewhat recently). Maybe it could be something more technical in your network settings preventing multiple connections outward? But it's been a long time since I had to mess with opening firewall ports/NAT/etc for online gaming. 

This is lovely and I'm sure your son really appreciates you wanting to spend time playing Battlefield with him.  

I just wanted to add that you're welcome to use my Portal playlists if you fancied some time with the two of you playing together on your own (some gadgets are problematic in solo/coop mode so Portal is sometimes better). I probably don't have all the maps in there, as there are some that I detest, but hopefully they'll be useful to you still. You'll just need to go to the "Host" part of the Portal menu and enter the codes. All of them have PVP AI so you have something to shoot at (it should keep you on the same side so long as you're in a party together). Feel free to @ me if you'd like any help with these or if you'd like the steps to build basic ones yourself. 

Conquest Large: absbwj
Conquest Small: aa7u6r
Breakthrough: abb11a
Rush: abcqf7

The current War Machine Rush Overide event is also PVE so that might be fun for you to play together. It's a squad of players against a swarm of robots.