After the new expansion, dragons are a regular pick in the draft. You can get around 7 dragons out of 30 on average, more if you go all-the-way and ignore better cards. Most dragons are awsome picks, so I am inclined to almost blindy select every dragon that is offered. Obviously, with curve etc. you shouldn't be too blindly selecting, but what is the downside? I've noticed a single counter to dragons in the shape of Dragonmaw Poacher, but that's the only counter available right? The Dragonslayer just rotated out, since Kobolds and Catacombs isnt' included anymore.
Other 'general' removal isn't too prevalent either right? That's based on just a few games though.
I would say that the dragons are pretty powerful, especially the new ones. Although I believe the most game-breaking dragon that you often see in arena now is Twin Tyrant. In the past couple of days at least 80% of my opponents have played it around turn 8-10 and many times they've had two copies of it. In most situations it's better than Flamestrike because it also gives you a 4/10 body. It's way worse than pre-nerfed Bonemare since it requires a minion to use its effect, while the Twin Tyrant can easily swing the game in your favor with almost no needed conditions. I honestly think that the card should be removed from arena or nerfed in some way, due to its ability of single-handedly winning you the game and the fact that every class has access to it, making it extremely common in arena.
As far as it goes for the other dragons, I think they're quite good. Well, with the exception of Dragonqueen Alexstrasza, which is beyond insane if you manage to pull off her battlecry. You could try picking up as many dragons as you can, so you can have as much dragon synergy as possible. Most of them are highly rated in HearthArena anyway, so if you're using the addon and always picking up the highest rated offered cards, you'll still end up with a lot of dragons.
Dragonmaw Poacher is obviously going to be a risk of getting faced against but I personally haven't seen too many people using it, so it's well worth the risk. As you mentioned, Dragonslayer is no longer available in arena so there isn't any other direct counter to dragons.