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.
Sorry to hear you are having wudu challenges.
While many people successfully complete wudu in Western clothing, I agree that Western clothing (especially formal wear for men) is more restrictive than some other cultures' forms of clothing, and may pose some logistical challenges (such as Sleeves That Do Not Rise or Inflexible Trousers), or is in an awkward space (Work Bathroom With Colleagues Staring). I am sure many of us have had Adventures in Clothing We Cannot Do Wudu In.
(This is setting aside other issues, such as not being able to undo buttons on sleeves due to a mobility problem, which would be a different sort of issue and would need a different intervention)
So, first, some general thoughts:
* If possible, try to wear clothing that you know you can do wudu in while wearing. I know that sounds simple, just putting it out there. Sometimes you can change what you wear, and sometimes you can't.
* If you are doing wudu in the same place regularly (such as a job site), perhaps you can keep some helper items around you, such as plastic slippers, or a stool. If it is awkward to keep those in a washroom area, maybe there are other locations you can do wudu, such as on an outdoor or park bench with a water bottle?
Sock Challenges vary according to madhhab.
* Sunni - The down side of Sunni-style wudu is that you have to wash your feet. Yes, it's awkward lifting up your feet and sticking them in the sink at work. Sometimes they even have rules against it. There may be other more comfortable or more discreet ways to accomplish this. You will have to think outside the box to find what works for you.
The plus side is, many Sunnis will allow wiping over leather socks as long as one has done wudu before wearing them. So, if you are doing Sunni-style wudu, you could see if that is a possibility. As in this article: https://islamqa.info/en/answers/9640/conditions-of-wiping-over-socks
* The plus side of Shi'i-style wudu is that you only have to wipe the feet. This can be done more discreetly without hauling your feet up into the sink, and is easier to do away from running water. Overally, Shi'i-style wudu is more water-efficient. But you do still have to wipe the feet, and there is no option to wipe over leather socks.
Some time ago, they used to sell "wudu socks" for women (I think made in Iran?). Perhaps they still exist. They were socks with flaps at the toes, so that, rather than taking off the entire sock, you could just open the flap and wipe the foot from the toe area, then close the flap back up. I am not sure if such items existed for men, but you could have a look and see if such a thing exists. If not, maybe make your own or have them made, or start your own product line?
Anyway, in short, yes, you do still have to do wudu as long as you actually have water and are able to perform wudu.
If you have a physical reason why you are absolutely unable to perform regular wudu (such as a cast on your foot), or some other sort of mobility consideration, that is a different situation and you should look up your specific case in accordance with the type of fiqh you follow.
But if it is just awkward, maybe you can think of some different ways to go about doing wudu that help to meet your needs better and make life easier. After all, Allah desires ease not difficulty!
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.