I am a big fan of Connemaras and halfbred (I am a bit big for most purebreds, but I have 2 halfbreds, on 14 h 3" and one 16h)
A lot is going to depend on the particular pony/horse.
The temperaments cover the whole spectrum, from “bombproof” to sensitive, from lazy to hyper-energetic.
The ones that are successful performance horses tend to be more on the sensitive side.
They are not USUALLY “spooky”, though I have seen some be VERY concerned about Paso Finos.
Like most pony breeds, they can be opinionated, but they can also be very accomodating.
Both of my halfbreds take a wide tree, and are comfortable to ride bareback, but they are not “round” like a draft horse.
They are almost all good jumpers, though not always with “good style” . Some are good at dressage (e.g. Seldom Scene) and others are more “pony gaited” .
They are mostly very healthy and sturdy, but there are a couple of things you want to watch for.
Like gray horses in general, gray Connemara are susceptible to melanomas. (one of my halfbreds has them, the other doesn’t).
There is also a genetic condition called Hoof Wall Separation Syndrome (aka HWSS) ( https://en.wikipedia.org/wiki/Hoof_w…ration_disease ) that occurs in Connemaras and halfbreds. Nowadays all registered Connemaras need to be tested, and all current breeding stock need to be tested in order for their offspring to be registered. But an older Connemara, or halfbred, that is not being used for breeding might not be tested. It is a relatively chap and easy (need a few mane hairs, complete with the “root”) at UC Davis.
But the take away is that if a Connemara or partbred has poor hoof walls, don’t just assume it is something that can be fixed with better shoeing. Check the HWSS status of the pony or its parents. One of my halfbreds is by a sire who is a HWSS carrier, but she is not affected. I have not had her tested.
Like other native pony breeds, they can be susceptible to founder from “rich” spring and fall grass, so some need to be muzzled or kept on a dry lot during critical times (I have not had to do that with either of my halfbreds).
Where are you located? Most Connemara breeders are happy to show of their stock, even if you are not at the point of actively shopping. Also, each region has a Connemara show each year, where you can see a variety of ponies and talk to people about them. And in Virginia, both the Upperville show and the Warrenton Pony shows have a Connemara division, where you can see a number in person.
Halfbreds are pretty great too.
