Question 1

Is this a real life thing? Wearing socks is mandatory for people washing dishes?

Yes, wearing socks is standard dress code policy in food services, regardless of the position maintained (in that scene Chris was actually dressing burgers).

Over the years I've worked at -- Papa John's, Jimmy John's, Starbucks, Fazoli's (twice), Pizza Hut, Dairy Queen, Steak N' Shake, and Outback Steakhouse -- and at all of them, they're sticklers for dress code.

In many cases, people are sent home if they violate dress code and can't fix the issue at the store (ex. I was sent home once for having a small hole in my jeans when delivering for Jimmy John's, maybe the size of a dime).

I've also seen people in get in trouble for not being clean shaven, not having slip-resistant shoes, having the wrong colored socks/shoes on, their shorts being too short, etc.. (I could go on and on)

If you're interested, here are general policies and procedures for Burger King dress code. Keep in mind that most Burger Kings are franchised (privately owned), so policy can vary slightly from one location to another.

Question 2

If so, why? what effect would it make if someone wears them or not.

It has to do with hygiene, health code and the general practices/standards/desired image of a company. From my experience, it's not necessarily default policy to wear socks, but if your ankles are showing, then it's mandatory. Since most of the time while working, when bending down, walking, etc., your ankles do show, most places go ahead and address this in dress code policy, instead of dealing with it so frequently on an individual basis.

Question 3

how did she manage to notice at all that he is not wearing socks? maybe the socks are very very short, you can't tell they are there?

It was pretty obvious that Chris wasn't wearing socks, since his pants were fairly short and his ankles visible. And, looking at his shoes, those probably violate dress code policy too; they're not black and slip resistant.

Question 4

what benefit this scene provides to the viewer? What are we meant to make out of it? What is the significance?

As the OP has pointed out, some people may consider these policies pointless/not a big deal. In Chris' case, he realizes that he's wasting his efforts trying to keep up with these small responsibilities, and that he should more actively be working towards his goal. So, he quits and moves on (starts hitchhiking).