Перед выходом из дома я проверяю, есть ли у меня студбилет.

Because перед in the meaning before takes the instrumental case.

• Dictionary form: выход
• Instrumental singular: выходом

So перед выходом literally means before the exit / before going out.

It is a noun here.

Russian often uses nouns where English uses a clause or an -ing form:

• перед выходом из дома = before leaving the house / before going out of the house

So instead of saying something like before I leave the house, Russian can naturally say before the leaving from the house.

Из means out of / from, and it requires the genitive case.

• Dictionary form: дом
• Genitive singular: дома

So:

• из дома = out of the house / from home

A useful detail: дома can also be an adverb meaning at home, but here it is not that adverb. Here it is the genitive form of дом after из.

Ли is a particle used to form an indirect yes/no question.

So:

• есть ли у меня студбилет = whether I have a student ID
• more literally: whether there is a student ID with me

The particle ли usually comes after the word that is being questioned. Here it follows есть, so the meaning is roughly whether there is.

Russian very often expresses possession with the pattern:

• у + genitive + есть + thing

So:

• у меня есть студбилет = I have a student ID

Literally, this is closer to:

• by me there is a student ID

That is the normal Russian way to say that someone has something. Russian does have verbs like иметь, but they are used much less often than English to have.

That is a very common learner question.

It is true that Russian often omits present-tense to be in sentences like:

• Он студент = He is a student

But есть is still used in existential and possessive patterns, especially when the idea is there is / there exists / do I have it available.

Here the speaker is checking for the presence of the student ID, so есть ли is very natural:

• я проверяю, есть ли у меня студбилет = I check whether I have my student ID

It is nominative singular.

In the structure у меня есть X, the thing possessed is grammatically the thing that exists, so it appears in the nominative:

• У меня есть студбилет

This is different from English, where student ID looks like a direct object after have.

A useful comparison:

• У меня есть студбилет = I have a student ID
• У меня нет студбилета = I do not have a student ID

Notice that with нет, Russian usually switches to the genitive: студбилета.

Проверяю is the imperfective present tense of проверять.

It fits because this sentence describes a habit or routine:

• Before leaving home, I check...

So the meaning is something like I regularly check.

If you used проверю, that would be perfective future:

• я проверю = I will check

That would refer to one completed checking action in the future, not a usual habit.

Студбилет is a common shortened form of студенческий билет.

It means student ID or student card.

This shortened form is very common in everyday speech and informal writing, so it is useful to learn even though the full form is more transparent.

Because that part is a subordinate clause.

The main clause is:

• я проверяю = I check

The subordinate clause is:

• есть ли у меня студбилет = whether I have my student ID

Russian normally separates such clauses with a comma, just as English often does with that, whether, if, and similar clause types.

Yes, Russian word order is flexible, and you could move parts of this sentence around. For example:

• Я перед выходом из дома проверяю, есть ли у меня студбилет.
• Я проверяю перед выходом из дома, есть ли у меня студбилет.

The original version puts перед выходом из дома first, which naturally sets the time frame right away:

• Before leaving home, I check...

So the original word order sounds very natural and emphasizes when this checking happens.