Sorry, I can't advise on breed as I might be a bit biased...
On a serious note, have more of a think of your normal day to day life, time, commitments you have and first look for a dog that fits your general lifestyle and then look at which breeds in that are suited for longer-distance walking taking into consideration there are some breeds with show lines and working lines as well with working lines usually having much greater stamina than show lines because they are bred to work, but they can come with a higher work drive than show or pet-bred lines. I'm obviously biased towards collies and there are a lot of them that end up in rescue because people don't consider the breed traits such as nipping and herding, the latter which if not tackled early can lead to issues with chasing cars, birds etc however if you have the time and commitment they are great hill companions, Moss and Zena easily do over 8 hours (before Moss was diagnosed with spondylosis) as did Millie before her.
If you go the rescue route, which I would recommend, the people there should be able to advise if they have a dog that is suitable for your lifestyle, it might not be a breed you have thought or a cross/mongrel that has the energy for a more active lifestyle so do take time to go home and research the breed(s) before making a decision.
In general, a dog should be between 1-2 years old before they start being introduced to hills to ensure their joints have time to develop properly, this is partly dependent on the size with smaller breeds able to start sooner collies fall in the middle, and large breeds need the longest time to finish developing properly. Even if getting a rescue that is the right age, no matter their energy, unless they have came from another hiking home then they will need building up to longer distances and terrain gradually, how long this will take will depend on the condition of the dog Moss came to me as a stray and had no muscle mass whatsoever and very soft paw pads so it was 6 months before she did her first Munro, Zena on the other hand I got direct from a farm at 14 months as a failed sheepdog so she was able to start hiking much quicker as she was in good condition to start with.
There is no guarantee a dog won't bolt after deer, training can reduce the chances of this but there is always a risk so it is knowing your dog and where you a walking. Zena will chase deer so where there are likely to be deer she is on the lead, Moss and Millie before her though were very reliable. Recall training is important if you want your dog off lead, but this isn't always possible - if there are sheep, steep drops etc dogs are safest on a lead - so I use a walking belt with a bungee lead for hands-free walking which is great, and handy for an extra pull up steep bits!
This post is looking a bit long, so I'll leave it there for now