După ce semnez, fac și o copie pentru Ana.

Romanian often drops subject pronouns when they are not needed, because the verb ending already shows who is doing the action.

• semnez = I sign
• fac = I make / I do

So După ce semnez, fac și o copie pentru Ana is perfectly natural without eu.
You could add eu for emphasis, but normally it is omitted.

După ce means after when it introduces a clause with a verb.

• După ce semnez = after I sign

This is different from simple după used before a noun or pronoun:

• după prânz = after lunch
• după el = after him

So here, because a full clause follows, Romanian uses după ce.

Because Romanian uses a conjugated verb after după ce.

• semnez = I sign
• a semna = to sign

English can sometimes use forms like after signing, but Romanian in this sentence uses a full clause:

• După ce semnez = After I sign

If you used a semna here, it would not fit this structure.

Semnez is the 1st person singular present tense of a semna (to sign).

So:

• eu semnez = I sign

Even though it is present tense in form, depending on context it can refer to:

• a habitual action: After I sign, I also make a copy for Ana
• a near-future planned action: After I sign, I’ll also make a copy for Ana

Romanian often uses the present tense this way.

Fac is the 1st person singular present tense of a face (to do / to make).

Here it means I make:

• fac o copie = I make a copy

This verb is very common and somewhat irregular, so it is worth memorizing:

• eu fac
• tu faci
• el/ea face

Here și means also / too / as well.

• fac și o copie pentru Ana = I also make a copy for Ana

It shows that making the copy is an additional action.
Without și, the sentence would simply say I make a copy for Ana, with less sense of in addition.

O is the indefinite article for a feminine singular noun, so:

• o copie = a copy

Romanian usually includes the indefinite article just as English does in this kind of sentence.

Compare:

• copie = copy as a bare noun
• o copie = a copy
• copia = the copy

Since the sentence means a copy, o copie is the correct form.

Pentru means for, and it marks the person who benefits from the action.

• pentru Ana = for Ana

This is the most direct and natural way to say that the copy is intended for her.

Romanian can sometimes express relationships through case endings, but in this sentence the simple preposition pentru is the normal choice.

The comma separates the introductory subordinate clause from the main clause.

• După ce semnez, = subordinate clause
• fac și o copie pentru Ana. = main clause

This is standard punctuation in Romanian when a clause like after I sign comes first.

The given word order is very natural, but Romanian is somewhat flexible.

The sentence starts with the time clause:

• După ce semnez, fac și o copie pentru Ana.

This emphasizes when the second action happens.
You could also place the main clause first in some contexts, but the original version is the most straightforward and common for this meaning.

Yes. A face o copie is a normal Romanian expression meaning to make a copy.

So:

• fac o copie = I make a copy

Depending on context, English might also say I make a photocopy or I copy it, but the Romanian phrase here is completely standard.