Hello,
I'll try to answer your questions in order.
1) You had correct conclusion Network > LAN > DNS only advertise these DNS servers to it's own DHCP clients, as in communication of your PC that is connected to router will be sent through AdGuard DNS, but communication of router will go through Mobile Service Provider (MSP) advertised DNS servers. This is done in case you need full access to everything for a router to fulfill it's functions, but you want to limit access to clients connected to router, be it yours employees or kids, or someone else.
2) In Network > Interfaces > Mob1s1a1 > Advanced settings if you would unmark Use DNS servers advertised by peer and enter custom DNS servers all communication going through this interface will use those DNS servers (except if you set different DNS servers in LAN settings for DHCP clients, their communication again will be forwarded through those servers).
As for connectivity problems AdGuard DNS depending on which servers you are using blocks certain IPs and websites from access, so it is natural that it might block some of yours connections. You can try their "Non-filtering" servers: 176.103.130.136 and 176.103.130.137 and see if it makes a difference, but other than that, if DNS servers that you chose to use are blocking yours connections, there is nothing else we can do from a router side, except not to use them.
3) Network > DNS configuration is for occasions, when you want to use router itself as DNS server. If that is not yours goal, do not change its configuration.
Hope This helps to understand how to configure DNS servers on RUTX routers.
Best regards,
VidasKac.