Tot discipulae adsunt ut magistra laeta sit.

Because discipulae is nominative plural: female students / girl students.

The adjective tot means so many or so numerous, and it is describing more than one student, so discipula has to be plural.

• singular: discipula = a female student
• plural: discipulae = female students

Here discipulae is also the subject of adsunt, so nominative plural is exactly what we expect.

Tot is an indeclinable word, which means it does not change its form.

So Latin uses:

• tot discipulae = so many female students
• tot pueri = so many boys
• tot libri = so many books

Even though the noun changes for gender, number, and case, tot stays tot.

Adsunt is from adsum, adesse, which means to be present, to be here, or to be in attendance.

It is made from:

• ad- = to, at
• sum = I am

So adsunt is the 3rd person plural present active indicative:

• adsum = I am present
• ades = you are present
• adest = he/she/it is present
• adsumus = we are present
• adestis = you all are present
• adsunt = they are present

Because the subject is plural (discipulae), the verb is plural too: adsunt.

Here ut introduces a result clause.

A very common Latin pattern is:

• tam / ita / sic / tot / tantus ... ut ...

This means:

• so ... that ...

So in this sentence:

• tot discipulae adsunt = so many female students are present
• ut magistra laeta sit = that the teacher is happy

The first part sets up the degree or amount, and ut introduces the result.

Because after ut in a result clause, Latin normally uses the subjunctive.

So:

• sit = present subjunctive of sum
• not est, which would be indicative

This is one of the most important things to recognize in Latin syntax:

• ut + subjunctive can introduce several kinds of clauses
• here, because of tot, it is a result clause

So ut magistra laeta sit means so that the teacher is happy / with the result that the teacher is happy.

It is a result clause, not a purpose clause.

The clue is tot. Words like these often signal result:

• tam = so
• ita / sic = so, thus
• tot = so many
• tantus = so great

A purpose clause usually expresses intention: in order that.
A result clause expresses what happens as a consequence: so ... that.

Here the idea is not that the students are present in order to make the teacher happy.
It is that so many are present that the teacher is happy.

Because laeta agrees with magistra.

• magistra is feminine singular
• so the adjective must also be feminine singular: laeta

Agreement in Latin adjectives follows the noun they describe in:

• gender
• number
• case

So:

• magister laetus = the male teacher is happy
• magistra laeta = the female teacher is happy

Magistra is nominative singular.

Even though it comes after ut, it is still the subject of sit:

• magistra ... sit = the teacher ... is

So inside the ut clause:

• magistra = subject
• laeta = predicate adjective describing the subject
• sit = verb

Latin often places the verb at or near the end of the clause, especially in formal or literary style. So:

• ut magistra laeta sit

is a very normal Latin word order.

That said, Latin word order is more flexible than English because the endings show the grammatical relationships. The same basic meaning could still be understood if the words were arranged differently, but this version is natural and standard.

Yes. Ut is a very common word and can have several meanings depending on context.

Some common uses are:

• ut + subjunctive for purpose
• ut + subjunctive for result
• ut meaning as
• ut meaning when in some contexts

In this sentence, the presence of tot strongly points to result, so ut means that in the sense of so ... that.

Yes. Sit is present subjunctive, and in this sentence it shows a result happening in the same general time frame as the main verb adsunt.

• adsunt = are present
• sit = may be / is, in the subjunctive after ut

In smoother English we usually do not translate it as may be here. We simply understand:

• so many students are present that the teacher is happy

So the tense is present, but the important grammatical feature is really the subjunctive mood.

Not in the same structure.

For a result clause after tot, Latin normally uses ut + subjunctive. So:

• tot discipulae adsunt ut ...

is the expected pattern.

Quod usually introduces things like:

• a fact clause: the fact that
• a causal clause: because

So quod would not work as a normal replacement here.

A good way is to split it into two parts:

1. Tot discipulae adsunt
   = main clause

2. ut magistra laeta sit
   = subordinate result clause

Then notice the signals:

• tot = so many
• ut
• subjunctive verb (sit)

That combination is a classic marker of result in Latin.