A shoe is an item of footwear intended to protect and comfort the human foot while the wearer is doing various activities. Shoes are also used as an item of decoration and fashion. The design of shoes has varied enormously through time and from culture to culture, with appearance originally being tied to function.
Yes it is permissible to do that, even you can keep Najis leather or any other Najis items in your pocket during Salah as far as Najis items are not wet because wet Najis item will make your pocket Najis which will invalidate your Salah.
If people take it as an insult or disrespect to religion, then it should avoided, but if it does not have such effect on you or others, then it will be permissible.
If there is a common understanding among people who attend the place that those who take shoes of others by mistake, allow using their own shoes by owners of the shoes which they have taken, the matter is solved.
Even if such agreement is not been declared in public but it is been understood as a matter of common sense, the matter is solved also.
Only if you know that the real owner of the left shoes (who is the person who has taken by mistake your shoes) does not allow using his shoes by you, then you need to seek his permission. This case is very rare.
Wudu can not be performed on shoes or socks. Wudu has a compulsory part which is wiping over part the head and on the feet. Feet must be uncovered while wiping on them.
If Wudu was not possible for any valid reason then Tayammum is to be performed in which we put our both hands on the soil or sand or stones etc, then wipe over our forehead from upper part of the forehead down to the end of it, then wipe over the outer part of the right hand then outer part of the left hand.