It Should Be A DC 10 Handle Animal
Here's the rules from Handle Animal:
This task involves commanding an animal to perform a task or trick
that it knows. If the animal is wounded or has taken any nonlethal
damage or ability score damage, the DC increases by 2. If your check
succeeds, the animal performs the task or trick on its next action.
So, does "squeeze into a cave" count as its own trick, or can you use an existing trick to do that? Ultimately that's the DMs call, but my interpretation is that it should be a DC 10 Handle, using the Heel trick:
The animal follows you closely, even to places where it normally
wouldn’t go.
Squeezing into a cave certainly is a place that a Horse normally wouldn't go, but:
- The Horse isn't being threatened by anything in the cave, as it's you doing the handling and it knows you.
- It's capable of moving to the location you want it to without taking damage. It sounds like it may not even need to make a check to squeeze, based on your description. (Checks aren't required if it has half the necessary space, so a one square wide tunnel in this case.)
- It also has taken no damage that you've mentioned
Based on those things, no special conditions apply here. Given that, I would rule this as a DC 10 using the Heel trick. (The Come trick has similar wording and would also work if you don't know Heel.)
If you don't have either of those tricks, then it gets a bit trickier. There's no rule on simply leading an animal where it doesn't want to go specifically, and in the absence of training there is a stronger case to require the DC 25 Push check.
Enforce the Handle Animal Rules
Getting those dogs to do what they want requires a Handle Animal check (DC 10 and a move action if it's a trick). Each PC can only do that to one animal at a time. Otherwise, the dogs will just generally do whatever you as the DM want them to. That might mean they all swarm something, or it might mean they find the target frightening and run away from it.
Remember, animals will only normally attack other animals, humanoids, monstrous humanoids, and giants. The animal has to be specially trained and use two tricks to have an attack command against anything else.
Kill The Dogs
Unless these are advanced animals with bonus HD, Dogs have 6HP. At 5th level, one Fireball will wipe them all out (the Riding Dogs you gave the party have 13 HP, so their survival chances are better but it should still kill some of them and make the Handle Animal checks harder on the others due to being wounded), and since Dogs are pack animals and one PC can't direct five dogs to attack five different things in one turn, they're likely to not be that far apart.
A Barbarian with Cleave could make short work of them as well. I'm not saying to go far out of your way to kill them, but they're low HP targets in combat, and they make a good target to even the numbers in a fight for certain types of enemies. If none of them are dying in combat at this level, it would be pretty strange.
You could pretty easily make an encounter that would eliminate the dogs in a turn or two, at which point the party will have to decide if it's really worth spending money to buy new ones.
Alternately, you could kill one or two of them in an encounter (which should happen normally in EL 5+ encounters), and see if the party decides that having dogs that aren't up to higher end combat around is a good idea. Any animal lover characters wouldn't want to put animals in harms way that aren't up to it, and the more frugal characters wouldn't want to have something die when they could sell it for gold and get more value out of it.
Initiative Shortcut
When this happens in my game, I roll one initiative for all of a given type (all the Dogs in this case), and have them all move at the same time. That cuts down the annoyance a little bit.
I also don't put much thought into what they're doing. Dogs are animals, not tacticians. They will either do exactly what they're ordered to do, or they'll attack, or they'll not attack. They're not going to get fancy figuring out how to reposition themselves for optimum protection of the PCs. (The PCs would have to use Handle Animal commands to do that.)
As they only have one attack, the rolling should be pretty limited and so you can keep the pace up by acting quickly without thinking about what they're doing.
Party Logistics, As You Mentioned
Is the party carrying enough food for five dogs? Are you enforcing that? Caring for five dogs in real life is actually quite a lot of work, make them deal with that.
Best Answer
There's no trick allowing a creature proficiency with armor
This reader is unaware of a handler being able to use the skill Handle Animal to teach a creature a trick that grants the creature actual proficiency with armor (but see below). For most mounts, this DM urges masterwork studded leather (PH 123, 126) (for a Medium humanoid 175 gp; 20 lbs.; adjust accordingly for bigger and littler nonhumanoids)—which has no armor check penalty—and not worrying about it.
However, you may want more than that.
The Dragon #292 (Feb. 2002) Wizards Workshop column "Sage Advice" includes the following exchange:
(The column is unsigned.) So far as this reader is aware, this information is never repeated anywhere in the entirety of the official Dungeons and Dragons, Third Edition corpus.
Keep in mind that trained war mount (also war-trained—q.v. the Ride skill use Fight with Warhorse) is itself a slippery and ill-defined term in the core rules. It takes the Rules of the Game Web column "All about Mounts (Part Two)" (Feb. 2005) to explain that the purpose combat riding (described in the skill Handle Animal) makes a creature into a trained war mount (or war-trained or trained for battle or whatever), although the column doesn't use that language exactly. (Also note that those columns are dismissed by some fans as unreliable, sometimes in part but sometimes as a whole.)
But neither animals wearing armor nor mounts wearing barding is addressed in the columns "Animals" (Parts 1–4) and "All about Mounts" (Parts 1–5) (these latter, in some cases, being the only source of some information about mounts). The core rules, as the question notes, aren't much help either. Indeed, it is a pretty miserable oversight that the core rules aren't clearer on this when the game has as one of its central classes a warrior astride a mighty steed.
To bring this all together, this DM recommends that the purpose combat riding gives the creature proficiency with light, medium, and heavy armors (yet probably not the associated feats—there are already enough uses for the power psychic reformation) and that part of the price of the warhorses and the war pony includes them being trained for the purpose combat riding.
The armor trick exists, but it doesn't do what it probably should
The Masters of the Wild (Feb. 2002) armor trick says, "The animal is willing to accept the burden of armor" (18), but it later explains that
Yay? So the armor trick is not a way for the animal to gain armor proficiency but a trick tax on the masters of unusual creatures. This DM recommends ignoring the armor trick completely: masters of unusual creatures typically have enough training issues already.