Hay más folios sobre la mesa de los que necesito, así que te doy la mitad.

Because hay is the usual way to say that something exists / there is / there are somewhere.

• Hay más folios sobre la mesa... = There are more sheets of paper on the table...

Spanish normally uses:

• hay to introduce the existence of something
• estar to say where something specific is

So:

• Hay folios sobre la mesa = There are some sheets of paper on the table.
• Los folios están sobre la mesa = The sheets of paper are on the table.

In your sentence, the speaker is introducing the existence of extra sheets, so hay is the natural choice.

In Spain, folio often means a sheet of paper, especially a plain sheet used for writing or printing.

A rough comparison:

• folio = sheet of paper
• hoja = sheet / leaf, more general
• papel / papeles = paper / papers, also more general

In Spain, folio is very common in everyday speech for loose sheets of paper. A learner from Latin American Spanish may hear hojas more often in similar contexts.

This is a very common Spanish comparison structure:

• más + noun + de lo(s) que...

It means more + noun + than I...

So:

• Hay más folios sobre la mesa de los que necesito = There are more sheets of paper on the table than I need

English uses than, but Spanish often uses de lo que / del que / de la que / de los que / de las que in this kind of sentence.

More examples:

• Tengo más dinero del que necesito = I have more money than I need.
• Compró más manzanas de las que quería = She bought more apples than she wanted.

Because que refers back to folios, and folios is masculine plural.

So the form has to agree:

• folio → masculine singular → del que
• folios → masculine plural → de los que
• casa → feminine singular → de la que
• casas → feminine plural → de las que

That is why we get:

• más folios de los que necesito

If there were no specific noun to agree with, Spanish could use de lo que instead:

• Es más de lo que necesito = It’s more than I need.

So here, de los que is used because there is a clear plural masculine noun: folios.

Normally, no. In standard Spanish, that sounds wrong or at least incomplete.

With a noun after más, Spanish usually needs the de los que / de las que / del que / de la que / de lo que pattern:

• Hay más folios de los que necesito ✔
• Tengo más ropa de la que uso ✔

If you say más folios que necesito, it does not sound like a normal comparison of quantity.

Because Spanish often leaves out subject pronouns when they are clear from the verb ending.

• necesito already means I need
• doy already means I give
• te doy means I give you

So:

• de los que necesito = than I need
• te doy la mitad = I give you half

You could say yo necesito or yo te doy, but it is usually unnecessary unless you want emphasis or contrast.

Te is the indirect object pronoun, meaning to you.

• doy = I give
• te doy = I give you
• te doy la mitad = I give you half

Spanish often uses these short object pronouns where English may use a separate phrase:

• Te doy un folio = I give you a sheet of paper
• Le doy un folio a María = I give María a sheet of paper

So te tells us who receives the half.

Because la mitad is the normal fixed expression for half as a noun.

• la mitad = half
• la mitad de los folios = half of the sheets
• te doy la mitad = I’ll give you half

Spanish usually uses the article here:

• la mitad
• el doble
• el resto

Also, mitad is a feminine noun, which is why it takes la.

It can be left unstated because the context makes it obvious.

The first part talks about folios, so when the speaker says te doy la mitad, Spanish naturally understands:

• la mitad de los folios
• half of the sheets of paper

Spanish often omits repeated information if it is already clear.

So these are all possible:

• Te doy la mitad = I’ll give you half.
• Te doy la mitad de los folios = I’ll give you half of the sheets.
• Te doy la mitad de ellos = I’ll give you half of them.

The version in your sentence is the most natural because it avoids unnecessary repetition.

Así que means so, therefore, or therefore/that’s why depending on context.

Here it connects the two ideas:

• There are more sheets than I need,
• so I give you half.

So:

• Hay más folios sobre la mesa de los que necesito, así que te doy la mitad.

It shows a logical result.

Other common connectors with similar meanings are:

• entonces = then / so
• por eso = that’s why / for that reason

But así que is very natural here.

Yes. Once la mitad is clear, it can be replaced by la.

For example:

• ...así que te doy la mitad.
• ...así que te la doy.

Both are possible.

The original sentence uses la mitad because it is clearer and more explicit. Te la doy is shorter and works when the listener already knows that la refers to la mitad.

Because each part attaches naturally to what comes before it:

• Hay más folios = there are more sheets
• sobre la mesa = on the table
• de los que necesito = than I need

So the sentence is built step by step:

1. what exists: más folios
2. where they are: sobre la mesa
3. how many in comparison to your needs: de los que necesito

Spanish word order is flexible, but this order is very natural and easy to process.