In the office, we have a weekly meeting every Monday morning.

This week is the last work week of the year, so we celebrated the anniversaries of some of our workeremployees, including myselfine.

In this sentences anniversaries are being celebrated, not people, so "mine" refers to your anniversary.
"Employees" fits with a white collar job in a business where individuals are being recognized for their contributions.
"Workers" is ok too, but it is often used to refer to blue collar jobs such as factory workers, service industry workers. It is not a strong demarkation, and some people may disagree with me.

The workerEmployees received a special item every fifth year, which wais carved with their names and honorary words on thea marble brick.

This happens every year. It's habitual, so the present tense is appropriate.

Today, I received the one.

If you remove "the" then it means you received a five year anniversary brick.
If you leave "the" it is confusing, and the reader may wonder if you received a one year anniversary brick. If that is what you received, then you should specify" "I received a one year anniversary / first year anniversary brick."

After that, each manager gave words to the anniversary workersthe managers spoke about their employees.¶
After that, each manager spoke about the employees who were recognized.¶
After that, each manager spoke about the employees whose anniversaries were being celebrated.

There are a number of grammar challenges here: If you start with each manager, you are forced to write his, her, his/her, or their. These options are awkward (first three) or questionable (their). To avoid these pitfalls, you can use the plural form.
Since you have already told your readers about the event, we can infer which employees the managers spoke about.

I also got a wonderful speech from my manager, and I couldn't control my tears.

"got" is not exactly the right word for this. I'm not sure if there is one, or I just can't think of it."Received" is a little better because it is more formal, like the event, but I don't really know if people "receive" a speech. It sounds a little odd to me. Another way to say this is:
"My manager made a wonderful speech about me..." English prefers the active voice!

This company is the second workplace I've worked since I moved to this country.¶
This company is the second one I've worked for since I moved to this country.

You could say "second workplace." Maybe that is more common in British English.

When I joined this company, my English was terrible (well, it still nowis), but everyone was really good to me.

Parentheses can be very effective. Good for you to challenge yourself to use them like this.

Their patience and kindness helped me to adoapt to the new workplace quickly and develop myselfcontribute.

"Adopt" and "adapt" are different. "Adapt to" meaning to "fit in, to get used to," works well here.
"the new place" this is a good spot for "workplace"!
The phrase "develop myself" isn't common or clear. It might be good to be specific "develop my skills" or change your meaning to "and contribute." If you are contributing, it shows you are keeping up which would include developing new skills, understandings, etc.

The workers received a special item every fifth year, which was carved with their names and honorary words on thea marble brick.

This might be nit-picking but "A marble brick" might be more suitable since you haven't mentioned it in a pervious statement.

Today, I received the one.

After that, each manager gave words to the workers that had anniversary workers.

Not fimiliar with the term "anniversary workers". So I kept it as describing word.

Their patience and kindness helped me to adopt the new place quickly and developwork on myself.

OR "help me with my development". I personally wouldn't use "develope myself" I haven't heard it being used that way often.