Lower Intermediate - Public toilet

Two questions, one is about (eash), could it mean (why)?
Second is about the image that came with the lesson, the word (مرحاض) is used there, I often hear it instead of (hammam), is there a difference between them?
Shokran in advance

I think you're thinking of ليش (laysh, which means "why")-note the "ysh" sound there- which can also be pronounced لي (lay), depending on where that person is from.

In the phrase "يسمح ل ", I thought it had to be have "ب", for example: "تسمح له باستخدام " or am I wrong? If not, when can I use "ب" in this type of phrase? !شكرا الدرس عظيم

(eash - إيش) can only mean (what) in colloquial Arabic, but I think what you meant was (leash - ليش) which means (why) in colloquial Arabic too.

The difference between (7ammam - حمام) and (mer7aa9’ - مرحاض) is minor. The word (7ammam - حمام) is more general and includes both the toilet and the bathroom, while (mer7aa9’ - مرحاض) is more specific for the toilet.

Afreeqiya;
Using تسمح له باستخدام is right, or you can use تسمح له بأن يستخدم is right too, where إستخدام and أن يستخدم can replace each other.

Notice in both cases above we used تسمح بـ , and this is the classical way of saying it. While in the lesson we have used mostly colloquial words or simplified classical, therefore we dropped بأن so it was تسمح يستخدم, in fact أن is rarely used in colloquial.

I'm afraid I've generally found café toilets in Arab countries disgustingly dirty, accept for the few employing female cleaners