I've spent hours on this and it seems I've solved my problem.
Now:
In pfSense web interface, it checks for updates without issues
for real (previously it said it was running the latest version but now it says that update is available).
In Ping menu it can ping all addresses without issues, e.g. 8.8.8.8, 192.168.2.1, google.com etc.
LAN clients can access the Internet.
What exactly fixed my problems (from what I understand):
In [System - Routing - Gateways] there were 3 gateways configured (2 first ones remained from my testing environment). I had no idea that (a) pfSense would save them all even after I've changed the main gateway IP in vWAN interface, and that (b) it would also still use the very first gateway as by default.
I've removed first 2 gateways, leaving only current one. I think this fixed the issues in Ping menu and returned the ability of pfSense to actually access and share Internet access.
In Hyper-V virtual switches settings, I've reverted back to vWAN with disabled "Allow management" and vLAN with enabled "Allow management".
Network Connections section in Windows now looks like this:
NIC1, set to something different from 1*.2*.3*.250, e.g. 1*.2*.3*.251 (also had to change the mask to allow that).
NIC2, with all protocols/components except "Hyper-V Extensible Virtual Switch" disabled.
vEthernet (LAN), set to use IP address 192.168.2.1, gateway is 192.168.2.2.
Did not change addresses in pfSense, but just in case here they are:
vWAN: 1*.2*.3*.250 (including all other parameters from our ISP)
vLAN: 192.168.2.2.
Important thing to note about (3) and (4) is probably that the physical server should not use the same IP addresses as pfSense uses on any interface. Otherwise, from my observation, LAN clients could ping all Internet addresses (e.g. 8.8.8.8 and google.com) but were unable to load any website for browsing.
Also, I'm not sure about DNS for now, but I've switched from DNS Resolver to DNS Forwarder. I was unable to fully understand how they work and why in most cases I'm unable to test DNS functionality right after I enable either service. Perhaps DNS services need some time to cache something, or I need to "Reset States" (have not tried this after changing DNS services yet). I'm testing DNS functionality like this in command line in Windows:
nslookup google.com 192.168.2.2
I'm disappointed that pfSense made me believe it could connect to the Internet at first. I might have been able to solve the problem much faster if it displayed an error message on Dashboard about being unable to check for updates.