Firstly, check out the link in my signature for one good place to learn to take decent hoof pics. And I suggest you do what you can to educate yourself as best you can on hoof function & form, as there are many incorrect assumptions & 'hoof ignorant' people out there, even among professionals, and there are many factors which come down to management/owner care, rather than just a 6-weekly farrier visit.
This friend told me, that after our regular farrier did Lucky's hooves they started to feel warm and she suspected, they were trimmed too short and she acquired an inflammation of the hoof-skin.
Hooves *generally*(it depends) SHOULD feel warm. If they feel really hot(when the horse isn't standing in direct sun), &/or the horse has a 'bounding' digital pulse, this may indicate laminitis - what your friend meant by 'inflammation of hoof-skin'. If being warm was the only reason for her to suspect a problem, then I wouldn't worry about it. But perhaps the horse is sore, shifting weight constantly, obviously over trimmed or some such, then there may be a problem. And yes, over trimming can indeed cause sensitivity &/or laminitis.
This is something that never happened before - not to my knowledge anyways, and no one ever told me something like this before.
As per my first paragraph above, that is because many have no/little clue about hoof health - UNhealthy, ILLfunctioning hooves are more the norm than truly healthy, strong ones, & people often don't even recognise that. So educate yourself, for the sake of your horse, don't wait for 'someone' to tell you something might be an issue.
just that her soles are rather thin on the front hooves and he would recommend to have her shod at least for a while. But only on the front hooves.
Our friend says, she should be shod on all 4 hooves,
Yes, thin soles are a common prob. It does tend to be a 'symptom' of the 'real' issues though, such as hoof form being a prob(you mentioned long toes for eg) If your vet was talking regular rim shoes, they will provide no protection/support under the foot and can cause further issues if hoof form is already a prob too. They are often an effective *palliative* though, if a horse is 'ouchy' on hard/rough footing or such though.
IF you use plastic shoes, or shoe with rubber pads, then there will be some sole protection. If you're already using hoof boots on rough ground, and the ground she lives on is yielding, if she is not sore or sensitive at home, then I imagine there will be no need for shoes.
Fixed shoes of any kind do come with some... less desirable effects, especially if hooves are unhealthy when shoes are applied. Sometimes it can be a 'necessary evil' & sometimes the 'side effects' may be extremely small, but it pays to educate yourself on the pros AND cons, so you can make more objective decisions.
also because she doesn't like the way her hooves grow (too flat, too short heels, too long toe). She also says, this is due to bad work of our regular farrier.
Short heels(esp compared to what is often seen) are good, as a rule. Of course, it's possible heels have been trimmed too short, or(rarely IME) overworn. But often, esp if toes are long/run forward, when people talk of 'too low heels', it is that heel walls may well be too long, but they've collapsed/crushed forward.
as long as Lucky doesn't get any problems (lameness etc), we should just keep her barefoot because with shoes she would need especially short trim intervals as her hooves would have NO natural abrasion at all and would grow longer even faster and they are already growing fast as it is.
I agree with this generally. Hooves are always best trimmed little & often, rather than only 6-8 weekly, as is the norm. Depends on many factors as to how often is best for any horse, but 3-5 weekly tends to be adequate. If a horse is shod, I would advise it was trimmed/shoes reset at least 5 weekly max.
I really doubt that she'll grow thicker soles just because we get her shod
Yes, correct. Using shoes will not cause her to grow thicker soles, and if anything, will compromise hoof function & health further and so cause soles to become even thinner. As said though, it depends & sometimes may be necessary, so, do your research.