Out of curiosity can they "talk" to each other if you use their IP address rather than their "name"?

I have some answers! What a tangled web...

So here are what I've found so far
* Kaspersky firewall has different settings around ICMP on both the desktop and Laptop. No idea why - just different combinations on each.
* Kaspersky firewall on the server has all ICMP settings at allow.
* Kaspersky firewall was blocking Remote Desktop on the desktop.
I've no idea how any of this came about. I have records and the only mention they make is of disabling the email protection on the server and disabling popup supression the clients because it slows browsing. However the server was upgraded from a previous firewall so Kaspersky might have inherited the settings from that?
But to complicate things further I'd forgotten that in an attempt to stop my Harmony Hub from disconnecting I'd temporarily set my DHCP lease to be several hundred days (it's back to 12 hours now). The desktop was okay because it's used so rarely that it's lease had been renewed. But I had two entries for my laptop in DHCP. Unfortunately it appears that the only way to clear these is to reboot the router and I can't be bothered. I don't need the desktop now so I'll shut it down and chances are that next time I actually want to talk to my laptop the old leases will have expired anyway.

I'm still puzzled why the desktop and laptop won't use IPv6 by default (with the changes to Kaspersky they both ping just fine if forced to). But maybe when the DHCP is sorted out IPv6 will come good as well.

Thanks for all your help, folks - several of your suggestions helped direct my thoughts toward the solution

Yes modern operating systems will use IPv6 by default, but adding anti-malware tools into the mix might change that, if they aren't fully compatible with dual stack (say if IPv6 was bolted on to an existing product rather than a first-class citizen).

The other thing is Happy Eyeballs

I really do not like Kaspersky, it's always had weird buggy issues. From my perspective there is little value in paying for an Antivirus nowadays, just my two cents.