I think most of the potential issues you list are non-issues, mostly because the technology required to augment people to the degree that you're suggesting solves those problems too. Let's take a closer look:
- People not changing their diet get fat(ter).
Simply augment them with an artificial stomach or other augment to speed up metabolism. Alternatively, you could force a change in diet by making people less hungry: after all, feelings like "hunger" are just chemicals. There's no rule that states you can't have an artificial gland to regulate one's sense of hunger.
- Lack of movement can cause heart failure. Increased further by possible fatness.
Besides obesity being engineered away (not difficult to do if you have augmentation tech), why not simply have artificial hearts? Furthermore, engineering a base biological fitness also wouldn't be to difficult. There are plenty of sci-fi solutions to making people who don't exercise fit--the reason that humans loose muscle mass when not exercising is because this has evolutionary advantages (a lot of biology is use-it-or-lose-it). Just as an example, you could implant thin wires inside biological muscles that artificially stimulate them and give them a "workout" while the person is asleep.
- Lack of movement can cause depression, tiredness and related mental problems.
Again. Depression, tiredness, and other mental problems are just chemicals in the brain. Exercise, for example, can release dopamine but we could also simply have an artificial organ which does this on command. Sure, solving mental health issues through copious use of drugs might not seem like an appropriate solution today, but in a future where these augmentations are ubiquitous, safe, and functional, I don't see why it wouldn't be possible to create a safe and effective cure for depression.
Additionally, the chemicals that are released into the brain through exercise can also be released in other human pursuits. Achieving a "feeling of accomplishment" is also possible through intellectual, creative, or social activities.
- No happiness or getting ‘high' from sport (runners high) when you do practice it.
See previous point. This can be solved with an artificial runner's high gland.
- Achievements like climbing a mountain can seem as ‘not your own'.
This is already an issue today, particularly in competitive environments where money directly translates to success. Today there's already great controversy about rich people who pay sherpas to "carry" them to the tip of Everest. The people that don't do this accuse those rich folks of "cheating".
I think that in a society where these augments are ubiquitous, physical achievement simply won't be very important to the people. Instead, skill-based accomplishments will be more highly regarded. For example, anyone can buy 'superlegs' but just because you have them, doesn't mean you're suddenly an expert surfer or rock-climber. Yes, you won't be bound by physical strength limits, but there's still a major skill component to the challenge.
- Body identity disorder or a form thereof could increase dramatically in augmented people.
This is an interesting point. Right now, BID is very rare, and mostly resigned to those who (often sexually) fetishize being an amputee of some sort. In fact, it's so rare that there is very little research on the topic or specific treatment outside of "standard" mental health treatment involving therapy and antidepressants.
Now, I'm not a medical professional, but I can tell you that the human brain has amazing neuroplasticity--the ability to adapt to extreme or unique change. This neuroplasticity is higher in children, but can probably also be induced through drugs. Regardless of how it's achieved, it allows users to adjust their own body image. Because of this, amputees with prostheses very quickly begin to see them as "part of myself". It's not uncommon for these people to begin feeling "phantom sensations", and that's with the relatively static and basic prosthesis we have today. I think the human mind would very quickly accept the augmented leg (or whatever), particularly if it moves and feels just like a real one.
Also, if someone's uncomfortable with their body, they can just get augments to change it.