Questions about example sentences with, and the definition and usage of "Izakaya"

A: I went to Izakaya and had a lot of yakiniku, which I had not had in a long time. It was really good! Today is the last day of a long time/week off, so I'm going to work tomorrow. I'll do my best.

A: × I stopped by the Izakaya after work overtime with my colleague.
✓ I stopped by the Izakaya after working overtime with my colleague.

× He knows how hard this job is because he is my subordinate who I trained, but he was exhausted of this overtime and was pale.
✓ He knows how hard this job is, because he is my subordinate who I trained, but he was exhausted by this overtime and was pale.

× So I treated him a healthy smoothie.
✓ So I treated him to a healthy smoothie.

× On the other hand, having not slept as many as 4 days, I drank some fruit juice and fell asleep while complaining about this overtime.
✓ On my part having not slept In as many as four days, I drank some fruit juice and fell asleep while complaining about this overtime.

× He complained that he couldn't sleep only 3 hours yesterday.
✓ He complained that he slept only three hours yesterday.

× The atmosphere of Izakaya was good and the darts were displayed.
✓ The atmosphere of Izakaya was lively, with darts displayed.

In written English we write outnumbers 1-10.

“On the other hand” is used only for contrasts, and is usually preceded by a contrasting phrase - which you don’t have here. As an example: on one hand I wanted to go to the party, on the other hand I had work I needed to get done that needed my time.

A: × Izakaya is a Japanese tyle bar / pub.
✓ Izakaya is a Japanese style bar / pub.

× Most of the Izakaya bars hopes customers to order at least 1 drink and 1 food per person.
✓ Most Izakaya bars hope customers order at least 1 item of food and 1 drink per person.

× In the most case, it is not a rule but kind of Japanese style manner staying Izakaya.
✓ In most cases, it is not a rule but kind of a Japanese style courtesy for eating at an Izakaya.

× Some Izakaya bars inform this rule on their menu or wall.
✓ At some Izakaya bars it is a rule, and inform of this on their menu or wall.

× This culture is diffrent and depends on the Izakaya.
✓ This culture can differ and depends on the Izakaya.

× I recommend this Izakaya because they kindly accepts customrers to order some foods without ordering drink (it is written on their official website), but when you go there at dinner time all custoer are chared cover chaege 300 yen.
✓ I recommend this Izakaya because they kindly accept customers ordering food without ordering drinks (it is written on their official website), but when you go there at dinner time, all customer are charged cover charge of 300 yen.

× Although, at the lunch time the charge is weaved.
✓ Although, at the lunch time the charge is waved.

× At dinner time, you can order main dish from a la cart menu and side dish such as rice type food and soup without a drink.
✓ At dinner time, you can order a main dish from an a la carte menu and side dishes such as rice and soup, without also needing to order a drink

This is pretty good overall, just a few notes.

Food is an uncountable noun, which means you cant have 1 food or 2 food, etc. but you can count an "item of food" or a "dish". It is also common to spell out smaller numbers (usually 1-10), so "one item of food and one drink"

a "manner" is a way of behaving, whereas a "courtesy" is a polite gesture and I think that fits a little better here. you could also say "It is not a rule, but is considered good manners, when eating at an Izakaya"

"Table charge" might be better for a restaurant. "cover charge" is the kind of term you'd expect at a nightclub.

"a la carte" is a French loanword often used in English and it refers to food that can be ordered as separate items, so "an a la carte menu"

A: ​‎What is an izakaya?

*An izakaya* is the equivalent of a pub in a Western country; it’s a place where hardworking office workers can stop in on their way home.
People drink, eat, and chat; *it’s really refreshing*. I really love to go to *izakayas* with friends from my office.
Izakayas have usually traditional decor and a paper lantern outside. Each izakaya has a different atmosphere and special dishes, *so I choose one depending on the purpose and my companions*.

____
It’s mostly perfect.

“Themselves” - this word is used when the subject and object are the same.
“It refreshes themselves.” X
“It refreshes them.” O
“They refresh themselves.” O
But, I think it’s more natural to say “It’s refreshing”.

A: Remove “and eat” from the end. It means the same thing. In Izakaya, it is common to share food.

A: "The izakaya was really crowded, but he was the only chef."