أستعمل كلمة المرور حتى أسجل الدخول.

Yes. Both أستعمل and أستخدم can mean I use in Modern Standard Arabic.

• أستعمل كلمة المرور = I use the password
• أستخدم كلمة المرور = also I use the password

In everyday modern usage, أستخدم is extremely common, especially in technology-related contexts. أستعمل is also correct and natural.

Because this is an iḍāfa construction, often called a construct phrase.

In an iḍāfa:

• the first noun usually does not take ال
• the second noun determines whether the whole phrase is definite

So:

• كلمة = word
• المرور = the passing / access / المرور

Together, كلمة المرور means the password.

Even though كلمة has no ال, the whole phrase is definite because المرور is definite.

Literally, it means something like word of access or word of passage.

But as a set expression in modern Arabic, it simply means password.

This is very common in technology vocabulary, just like:

• اسم المستخدم = username
• تسجيل الدخول = login / sign-in

Not here. In this sentence, حتى means so that or in order to.

So:

• حتى أسجل الدخول = so that I log in
• more natural English: to log in

Arabic حتى can mean different things depending on context. Two common meanings are:

• until
• so that / in order to

In your sentence, it expresses purpose, not time.

Because Arabic does not use an infinitive in the same simple way English does.

English says:

• to log in

Arabic often expresses that idea with:

• حتى + present verb
• لكي + present verb
• لِـ + verbal noun or لِـ + present verb in some contexts

So حتى أسجل الدخول is a normal Arabic way to say to log in or so that I can log in.

Yes. After حتى when it means so that / in order to, the following imperfect verb is normally in the subjunctive.

The fully vowelled form would be:

• أُسَجِّلَ

But in normal Arabic writing, short vowels are usually not written, so you just see:

• أسجل

That means the subjunctive is there grammatically, even though it is not visible in ordinary spelling.

For comparison:

• indicative: أُسَجِّلُ
• subjunctive: أُسَجِّلَ

Without vowel marks, both look like أسجل.

It is a standard expression meaning I log in or I sign in.

Literally, it is related to registering entry/access, but in computer and website language it is just the usual way to say log in.

Related forms:

• تسجيل الدخول = logging in / login
• أسجل الدخول = I log in
• سجّل الدخول = log in! / he logged in, depending on context and vowels
• تسجيل الخروج = logging out / logout

So this is best learned as a set phrase.

Yes, كلمة is grammatically feminine.

That matters when adjectives or pronouns refer to it. For example:

• كلمة المرور الصحيحة = the correct password
• نسيتُ كلمة المرور الخاصة بي = I forgot my password

In your sentence, you do not see much effect from that femininity, because there is no adjective agreeing with كلمة المرور.

Yes. A few very natural alternatives are:

• أستخدم كلمة المرور لكي أسجل الدخول.
• أستخدم كلمة المرور لتسجيل الدخول.
• أستعمل كلمة المرور لتسجيل الدخول.

These all mean roughly the same thing.

A small nuance:

• حتى أسجل الدخول = so that I log in / in order that I log in
• لكي أسجل الدخول = in order to log in
• لتسجيل الدخول = for logging in / to log in

All are standard, and the version with لتسجيل الدخول is especially common in formal or technical language.

Usually it sounds like a general present statement or a statement of purpose.

So it can mean something like:

• I use the password to log in
• I am using the password so that I can log in

The first verb أستعمل is present tense, and the second part حتى أسجل الدخول gives the goal or purpose of that action.

So the second action is often understood as something intended or expected to happen after the first.