Jag blev också överraskad, men inte heller barnen hade glömt presenten.

Because bli + adjective/participle often expresses a change of state.

• Jag blev överraskad = I got surprised / I became surprised
• Jag var överraskad = I was surprised

So blev focuses more on the moment or event of becoming surprised, while var describes the state itself. In this sentence, blev is the natural choice because it sounds like a reaction that happened.

In a normal Swedish main clause, if the subject comes first, the finite verb usually comes second, and adverbs like också often come after that verb.

So:

• Jag = subject
• blev = finite verb
• också = adverb
• överraskad = complement

That gives:

Jag blev också överraskad.

This is the neutral word order.

If you move också to the front, you change the emphasis:

• Också jag blev överraskad = I too was surprised

That is possible, but it is more marked and emphasizes I.

Inte heller is the negative counterpart of också.

A very useful basic contrast is:

• också = also / too
• inte heller = not ... either / neither

So in this sentence:

• Jag blev också överraskad = I was surprised too
• men inte heller barnen hade glömt presenten = but the children hadn’t forgotten the present either / nor had the children forgotten the present

English often uses either in a negative sentence, while Swedish commonly uses inte heller.

Yes, barnen is still the subject.

Here, inte heller is closely connected to barnen, so the whole beginning can be understood as a single fronted idea:

inte heller barnen = not the children either / neither the children

Because Swedish is a V2 language in main clauses, the finite verb must come in second position. If inte heller barnen is treated as the first constituent, then hade naturally comes next:

• Inte heller barnen = first constituent
• hade = finite verb in second position

So the structure is normal Swedish word order.

A more neutral alternative would be:

Men barnen hade inte heller glömt presenten.

That version is also correct, but the focus is slightly different.
The original sentence gives stronger emphasis to the children as another group included in the statement.

Hade glömt is the past perfect (also called the pluperfect).

It is used for something that was already true before another point in the past.

Here the past point is the speaker’s surprise:

• Jag blev också överraskad = I was surprised
• barnen hade glömt / inte glömt ... = by that past moment, the children had / had not forgotten ...

So hade glömt places the forgetting relative to that earlier past situation.

Compare:

• glömde = forgot
• har glömt = has/have forgotten
• hade glömt = had forgotten

Also note:

• infinitive: glömma
• supine/participle used in the perfect tenses: glömt

Because they mean the children and the present.

In Swedish, definiteness is usually built into the noun itself:

• barn = child / children
• barnen = the children
• present = a present, a gift
• presenten = the present

So instead of using a separate word like English the, Swedish often adds a definite ending:

• -en
• -et
• -na
• -en (plural, depending on the noun class)

In this sentence, both refer to something specific and known from the context, so the definite forms are used.

A useful detail:

• barn has the same form in singular and plural when indefinite
• barnen is definite plural

It comes from a verb, but here it functions like an adjective/past participle.

The verb is:

• överraska = to surprise

The participle is:

• överraskad = surprised

In Jag blev också överraskad, the finite verb is blev.
Överraskad describes the result or state of the subject.

So the sentence is built like:

• blev = became
• överraskad = surprised

This kind of structure is very common in Swedish:

• Jag blev trött = I got tired
• Hon blev arg = She got angry
• Vi blev förvånade = We got surprised

And yes, the form can agree in number/gender:

• Jag blev överraskad
• Det blev överraskat
• Barnen blev överraskade

This is a very common learner question, because English often uses too, also, and either in ways that do not map perfectly onto Swedish.

A simple rule is:

• use också in positive statements
• use heller in negative statements, usually with inte

Examples:

• Jag kommer också. = I’m coming too.
• Jag kommer inte heller. = I’m not coming either.

So in your sentence, the first clause is positive:

• Jag blev också överraskad

and the second clause is negative in meaning:

• inte heller barnen hade glömt presenten

That is why Swedish uses också in one clause and inte heller in the other.