Not to nit pick but a short rest after every fight isn’t typical. A short rest after every 2 fights maybe, but doubtful. My game is a short rest every 4-6 fights. What you are describing is a nova, just say so.
I have never seen or been in a game where you can take your warhorse into a dungeon, let alone have it live past the first or second encounter. Your horse has an average of 19 HP and various things can get you off it so you cant take the damage for it. I don’t see how it survives the first area attack in good shape at your level (6). I guess your group fails most group stealth checks no?
A troll can fit into many places a horse can’t, it’s shape is more malleable then a horse, but if your DM allows all the above I would have played a Paladin instead.
A Paladin gets an intelligent mount, so it acts on its own turns and you can communicate telepathically with it. Therefore you can prone them your shield then have your warhorse attack them with advantage for 2d6+4. In addition a Paladin can cast a spell with self as range and it affects both, so a Paladin can cast a smite spell on himself and can have his horse use the smite spell also. A Paladin on his mount is a holy terror.
A warhorse has these attacks also, RAW a PC can’t direct it to attack as it only has 3 action options and attack isn’t one. I would change that if you took the mounted combat feat. I would definitely talk to your DM about it as you have made a substantial investment in mounted combat.
The lance as a dueling weapon I consider an exploit, on horseback it is used across the second arm or rested on a shield in addition to being cradled under the arm. It requires Dueling is intended to reflect precision, a weapon condition that require you to be seated on a mount to me isn’t as intended. I would definitely not allow dueling there, that is IMO not RAI. But then I would also not allow a horse into a dungeon in most cases either, or would just have an intelligent enemy immediately try to dismount you or attack the horse, as most intelligent creatures would. Or set their reach weapons to receive a charge. I will ask Crawford though for what it’s worth about the lance though.
Edit- I checked the lance discussion and the rules for it RAW are absurd. RAW a halfling riding a dog can use a lance for D12 as a weapon since the lance doesn’t have the heavy property. With the dual wielder feat the same halfling could use 2 lances while mounted and attack with both from the back of his riding dog.
Yes there is a problem here. If your DM allows it though go for it!
A few things...
obviously campaigns differ. they have different degrees of intensity and attention to detail etc. they have different pacings. So as far as some of those above...
Short rest every fight - i can certainly see a campaign style where short rests or even long rests occur after **nearly** every fight if not all (whether or not that counts as "known" to the PCs or not.) It also depends on how one differentiates a single fight from an extended encounter. A classic cyberpunk style campaign is the "Blue Light Special" where the PCs are emergency rescue teams for eithe rhigh paying clients or part of a government outfit. They get a "code red" from a high price client, jet into the fray, extract their client and get back to base. That sets up the vast majority of their outings as one-and-done.
In such a campaign there would be obviously some times things get more complicated but for the vast majority of cases where they aren't the Gm would adjust his encounter balance efforts to take into account the "fully loaded - no hold back" aspects.
So, every one or every two or every three or two per long rest in a set of 6-8 are all just baselines that provide a consitent benchmark to help with the balancing - not right or wrong or RAW or non-RAW. Campaigns and groups define the game, not the other way around.
Mounts, warhorses, pets and other considerations...
Just two sessions ago in the weekly game i play in our group had our horses killed and our wagons stolen when four "gruesomes" (my PCs named for this homebrew monster type of my Gm) rolled into our caravan in thew hills and rolled us up pretty solidly so that we barely made it out with the gear on our backs and most of the PCs. they went for the horses first, since they were big enough to pull the wagons and that kept us from riding any of the wagons away. (For all of 5e's style of not so nitpicky about logistics (see spell component pouch and other examples) they really missed the boat by not including a first level equivalent of a "mount spell" for something as simple as routine travel needs.)
other campaigns i have seen "we tie the horse up outside the delve" and even days later when we come back out of the dungeon the horses are still there and we can begin the trek thru wandering monsters back to the city.
I once started in a supers campaign where the session started at the "scene of the criris" and (as it was explained to me) "that roleplaying part of *sitting in your secret ID at work and how fast do you get here when the alarm* we skip that now to save time" and that is not so much different from older days when "session starts at dungeon door" was not all that rare a bird. i did not stay with that particular supers campaign after the one session as it wasn't what i was after, but they had a blast and it lasted quite a while if i recall.
mechanically speaking...
lance
pro - 1d12 damage + reach
con - two handed unless nounted, disadvantage within 5' (tyhe most common melee range)
greataxe
pro 1d12 damage
con - heavy and two handed.
scratching my head to see how those are particularly any sort of majorly screwed between each other balance-wise.
Quick scan of mounted combat - dont see any special "add to lance" rules.
So, those two do not seem to me to be tragically imbalanced with each other. Do not see why a halfling wouldn't have access to a lance that can be used.
So maybe there is a problem with some class features causing whatever imbalance is believed to exist or a feat causing the imbalance but thats a horse of another color.
