A lot of this is seen much more clearly in Doom's Day.
There is no problem with priests linking from the Abyss to someone appropriate on the material planes etc.
The problem is in "having a link to the right person." Priests have links to avatars, typically saints, but sometimes archons (although official routings are to go priest->saint->archon, there is nothing stopping a direct connection to an archon if both parties allow). So if an avatar were on the material planes at the same time as a priest linked to him/her was in the Abyss, they could form a link (automatically) and in fact a bridge all the way through to the Outer Planes.
The problem for the Inferno was that none of the priests on the Inferno had established links with any of the saints/avatars that were either in Astlan or Nysegard.
In Book III, the majority of avatars on material planes were from Nysegard, and none of the priests on the Inferno were from Nysegard and so had no connections.
Of Tiernon's Astlanian avatars, you basically just had Hilda, and unfortunately, she's not that major of a saint. She did have a few adherents in the Rod, such a Danyel and there were others, but they were all soldiers not priests. She was their 'patron saint' but that link isn't as strong as a priestly link. I.e. he can't draw power on it, but he can pray to her, although it's likely this connection runs through her HALO (which would be in the Outer Planes and so not connected---but maybe it would fall back to her)
We can assume, since no one noticed their link to her come alive, none of the priests on the Inferno were priests linked to Hilda, otherwise they'd have had their full mana stream and they definitely would have noticed that and alerted people.
But yes, if a deity wanted to invade the Abyss, they could definitely station avatars on the material planes and route power to priests in the Abyss, or other avatars that way.
It is very doable and being done, and being done by people we all know in the books. The Outpost of the Nyjyr Ennead in The Abyss has links back to the Astlanian moon Uropia where their base is and then back to their remaining locations in the Outer Planes where their god pools are.
Thus Bess and her pantheon can have full access to their divine powers in The Abyss. But you did not hear this from me.
One reason that Sentir and Aodh meet with Lilith in Etterdam is because of this reason. They will go and have been there, but they really don't want to 1) they don't want anyone on material planes tracing them to the Abyss and 2) they can't draw on the full resources of their gods without someone doing a relay on the material, and that would be very obvious. So, they prefer to meet on more neutral ground where everyone has their full powers and it's far less suspicious.
And yes, the Doomalogue gates are very tightly warded against such activities. Specifically, the "World/Star/Doom Gates" which are little wormholes between realities scramble/break links.
When Tom connects to Doom through the Nysegard Doomalogue, he is not using the Doomgate. Rather he is using the other private connections between the Doomaluges that are highly encrypted links accessible only to Orcus and those linked to him. These "other links" are the same ones that were used to 'trickle charge' the Doom of Nysegard so that it could be restarted remotely.
The Inferno can and is about to leave the Abyss using and "Abyssal Gateway" (which is typically a spell/discipline rather than a permanent device/artifact) It's the circular ring of fire that rips a hole in reality that Tom likes to use to take material objects physically into the Abyss. That is how they got there, and while the ship was damaged, most of that damage was to the gravitational drive system, it was not quite as bad for the Abyssal Gateway Projector (there are 2, one in each of the two conning like towers at each end of the ship and links to the main power source--it was these links that were damaged to both projectors), however they had plenty of time to repair the broken links before they were ready to leave at the end of AoD.
Fortunately for them, the Oorstemothians tend to prepare for every possible situation, so they had plenty of repair materials, at least for the first round of battle. We shall see about the next, you can only pack so many spare parts in the ship.