Ni renkontiĝos la dekan de julio ĉe la biblioteko.

It breaks down like this:

• renkont- = the idea of meeting/encountering
• -iĝ- = makes the verb intransitive here, giving the sense of come together / meet each other
• -os = future tense

So Ni renkontiĝos means we will meet or we will meet each other.

Because renkonti is usually a transitive verb: you meet or encounter someone/something.

• Ni renkontos la instruiston = We will meet the teacher.

But when the people in the subject are meeting one another, Esperanto very often uses renkontiĝi:

• Ni renkontiĝos = We will meet each other / We will meet up.

Using renkontiĝos is the clearest and most natural choice here.

That -n is the accusative used for a time expression.

In Esperanto, a specific date can be expressed without a preposition by using the accusative:

• la dekan de julio = on the tenth of July

So the -n here does not mean a direct object. It marks time when.

You can think of it as a shortened form of something like la dekan tagon de julio = the tenth day of July.

Because julio is part of a de phrase:

• de julio = of July

The accusative is on dekan, the date expression itself. The month name stays after de, so it does not take -n here.

That is the normal way to say dates with ordinal numbers in Esperanto.

• la unuan de majo
• la dekan de julio
• la dudek-kvinan de decembro

This works like English the tenth of July. The article is part of the standard date expression.

Because Esperanto often uses the accusative instead of a preposition for dates and other time expressions.

So English on July 10 becomes:

• la dekan de julio

This is very common and completely normal Esperanto.

ĉe usually means at, by, or near a place.

So ĉe la biblioteko means at the library as a meeting location. It does not necessarily mean inside the building.

Compare:

• ĉe la biblioteko = at the library / by the library
• en la biblioteko = in the library

If you want to emphasize that the meeting is inside, en would be more specific.

In Esperanto, names of months and days of the week are normally written with lowercase letters unless they begin a sentence.

So:

• julio
• lundo
• decembro

This is different from English, where July must be capitalized.

Not here. renkontiĝi already gives that mutual meaning.

So:

• Ni renkontiĝos already means we will meet each other

If you used the transitive verb renkonti, then you might need extra wording for clarity, such as unu la alian in some contexts. But with renkontiĝi, that is unnecessary.

Yes. That is very common.

• la dekan de julio
• la 10-an de julio

Both are correct. The second is just the numeric way of writing the same date.