This question is extremely subjective, so anyone with experience will (and should) come up with his one personal way. What I will describe is therefore just an idea of when and why to use shoes with ankle support.

The distinctions between the two types that cause the different application exceed the (no) ankle support:

ankle support

• stiff sole
• heavy

no ankle support

• flexible
• lightweight
• generally more breathable

So whenever walking on easy terrain with moderate pack, use shoes without ankle support. The walking is more ergonomic and they are lighter.
On the other hand whenever technical stuff like obviously any mountaineering, rock scrambling, walking extensively on loose ground (scree), on snow/ice and when walking with heavy load you certainly want ankle support. Here in the first cases the chance of a misstep and in the second one the consequences are too bad and ankle support ways out the disadvantages by far.

Where it gets tricky is in between, the most typical case is multipitch climbing in alpine terrain where you have to carry anything (so your shoes) up the route. You usually have quite some gear to carry (but not like tons) and the approach to the climb is somewhat rough but not technical. While ankle support certainly wouldn't hurt the additional weight and size are a pain. To judge what to take in such a situation requires either knowledge of the path you will be walking or experience. And the conclusion will be that there is not one, or two, maybe not even three shoes to fit all scenarios. The more you are out there the more justifiable it will be to have several shoes: A super stiff, heavy shoes with ankle support for technical stuff, a lightweight shoes with ankle support and light trekking shoes do the trick for me at the moment. Even though I have to admit, the choice between he latter to still remains tricky :)

As to the weakening of ankle muscles in shoes with ankle support:
It is just common sense. If you constantly wear boots with ankle support, your ankle muscles will adapt. Thats another reason to wear light shoes too. Or even better: Do an occasional bare feet walk, this will improve many stabilisation muscles.

This question is extremely subjective, so anyone with experience will (and should) come up with his one personal way. What I will describe is therefore just an idea of when and why to use shoes with ankle support.

The distinctions between the two types that cause the different application exceed the (no) ankle support:

boots (ankle support)

• stiff sole
• prevents dirt and small rocks from getting in
• heavy

shoes (no ankle support)

• flexible
• lightweight
• generally more breathable

So whenever walking on easy terrain with moderate pack, use shoes . The walking is more ergonomic and they are lighter.
On the other hand whenever technical stuff like obviously any mountaineering, rock scrambling, walking extensively on loose ground (scree) or on snow/ice and when walking with heavy load you certainly want boots. Here in the first cases the chance of a misstep and in the second one the consequences are too bad and ankle support ways out the disadvantages by far.

Where it gets tricky is in between, the most typical case is multipitch climbing in alpine terrain where you have to carry anything (so your shoes/boots) up the route. You usually have quite some gear to carry (but not like tons) and the approach to the climb is somewhat rough but not technical. While ankle support certainly wouldn't hurt the additional weight and size are a pain while climbing. To judge what to take in such a situation requires either knowledge of the path you will be walking or experience. And the conclusion will be that there is not one, or two, maybe not even three models to fit all scenarios. The more you are out there the more justifiable it will be to have several models: Super stiff, heavy boots for technical stuff, lightweight boots and light trekking shoes do the trick for me at the moment. Even though I have to admit, the choice between he latter two still remains tricky :)

As to the weakening of ankle muscles in shoes with ankle support:
This is just common sense. If you constantly wear boots with ankle support, your ankle muscles will adapt. Thats another reason to wear light shoes too. Or even better: Do an occasional bare feet walk, this will improve many stabilisation muscles.