Hi there, to answer your questions:
1.- Can the employer simply skip the payment this month
Legally that may not be the correct thing to do but there is nothing topping them from doing that. Whilst you are able to challenge this as non-payment of wages and pursue them for what they owe you, by the time a claim is made and resolved, they would have likely paid you already as it will take longer than a month. You cannot force them to pay you sooner and taking legal action will take much longer than that so in reality they can delay payment even if it is wrong to do so in the circumstances.
2.- Can the employer fire me next week before June 23 because in the UK the companies can fire people before the first month on employment,without paying them anything?
They can fire you in the first 2 years of employment for no reason and without following a fair procedure. But they still need to pay you for the work you have already carried out, so it is not true that they can fire you and pay you nothing - they still have to pay you for the time you have worked for them so you cn pursue them for that money even if they dismiss you.
3.- I will resign on June 24th, when I will not receive the due payment in my bank account
ok you are free to do so at any time
4.- After my resignation, can I demand from the employer to be paid because of my month of work, and because of the Hotel expenses, and all the expenses I had to fund during my work?
You can indeed request to be paid for the work you have been doing but you will not be able to claim expenses unless you were entitled to them under contract. If the contract did not say you can claim expenses then you cannot pursue them for these