انا مليت الاستمارة، وبعدها مضيت عليها.

Yes. In Egyptian Arabic, the verb often already tells you who the subject is, so انا is not strictly necessary.

You could also say:

مليت الاستمارة، وبعدها مضيت عليها.

That would still mean the same thing.
Adding انا can make the sentence a little clearer or more emphatic, especially in speech.

In normal Arabic writing, short vowels are usually not written, so forms like مليت can look ambiguous by themselves.

Here, انا makes it clear that the speaker means I filled.
Without انا, context would usually tell you.

So in this sentence:

• انا مليت = I filled
• if someone said it directly to a man, مليت could also be understood as you filled

مليت is the past-tense verb meaning I filled or I filled out.

The basic Egyptian verb is ملا = to fill.
When you put it in the past tense for I, you get مليت.

So:

• ملا = he filled
• مليت = I filled

In this sentence, because the object is الاستمارة = the form, the meaning becomes I filled out the form.

الاستمارة means the form, while استمارة means a form.

Since the sentence is talking about a specific form, Arabic uses the definite article الـ.

So:

• استمارة = a form
• الاستمارة = the form

وبعدها is made of:

• و = and
• بعدها = after that / after it

Together, وبعدها means and after that, and then, or afterwards.

In natural English, you would usually translate it simply as then or after that.

Because بعد by itself usually needs something after it.

For example:

• بعد الشغل = after work
• بعد كده = after that
• بعدها = after it / after that / afterwards

So بعدها works as a complete expression here, while بعد alone would feel unfinished.

Because with the meaning to sign, Arabic normally uses the preposition على, literally on.

So the structure is:

• مضيت على الاستمارة = I signed the form
• مضيت عليها = I signed it

This is very natural in Arabic, even though English usually says just sign it without a preposition.

So literally, the Arabic is closer to I signed on it.

Because الاستمارة is a feminine noun.

The pronoun in عليها refers back to الاستمارة, so it has to match it:

• عليه = on him / on it for a masculine noun
• عليها = on her / on it for a feminine noun

Since استمارة is feminine, you say عليها.

Here مضيت means I signed.

Yes, this is a valid and natural way to say it. The verb is connected with على, so:

• مضيت عليها = I signed it

That said, in everyday Egyptian, many people also say:

• وقعت عليها

which also means I signed it.

So both are useful to know:

• مضيت عليها = I signed it
• وقعت عليها = I signed it

Yes. In Egyptian Arabic, all of these can work in similar contexts.

For example:

• انا مليت الاستمارة، وبعدها مضيت عليها.
• انا مليت الاستمارة، وبعدين مضيت عليها.
• انا مليت الاستمارة، وبعد كده مضيت عليها.

All of them can mean I filled out the form, then I signed it.

The differences are mostly about style and rhythm:

• وبعدها = a bit neater, sometimes slightly more formal
• بعدين = very common in speech
• بعد كده = also very common and conversational

A rough pronunciation is:

ana maleit el-estemāra, w baʿdaha maḍeit ʿalēha

A few notes:

• انا = ana
• مليت = roughly maleit
• الاستمارة = el-estemāra
• وبعدها = w baʿdaha
• مضيت = maḍeit
• عليها = ʿalēha

You do not need to pronounce it in an overly formal way. In natural Egyptian speech, the sentence flows pretty smoothly as one unit.