Después de leer la revista, mi hermana la tira a la papelera.

Because in Spanish, después de is normally followed by a noun or an infinitive.

• después de la cena = after dinner
• después de leer = after reading

So leer stays in the infinitive form because it comes after the preposition de.

A structure like después leer is incorrect here, and después que... is a different pattern that usually introduces a full clause, for example después de que mi hermana lee/leyó..., though that is not what this sentence is doing.

After a preposition like de, Spanish normally uses the infinitive, not the gerund and not a conjugated verb.

So:

• después de leer = after reading
• not después de leyendo
• not después de leyó

If you want a full clause with a subject, then you need something different, such as después de que leyó la revista. But when the idea is simply after doing something, Spanish prefers después de + infinitive.

That la is a direct object pronoun, and it refers back to la revista.

So:

• la revista = the magazine
• la = it

The second part literally works like:

• mi hermana la tira a la papelera = my sister throws it in the bin

Spanish often uses an object pronoun instead of repeating the noun again.

With a normal conjugated verb, object pronouns usually go before the verb:

• la tira
• lo compra
• las lee

They are attached to the end only in certain cases, especially with:

• infinitives: tirarla
• gerunds: tirándola
• affirmative commands: tírala

So in this sentence, la tira is the normal position.

You can say that, and it is perfectly grammatical.

But Spanish often avoids repeating a noun that has just been mentioned. Since la revista already appears in después de leer la revista, the next clause naturally uses la to mean it.

So both are possible:

• Después de leer la revista, mi hermana la tira a la papelera.
• Después de leer la revista, mi hermana tira la revista a la papelera.

The version with la sounds more natural because it avoids repetition.

Because tirar often expresses movement toward / into somewhere, so Spanish commonly uses a.

• tirar algo a la basura
• tirar algo a la papelera

This is similar to the idea of throw something into the bin.

Using en can sometimes appear in other contexts, but with tirar, a is very common because it marks the destination of the action.

Because papelera is a feminine noun:

• la papelera

The contraction al only happens with a + el:

• a + el parque = al parque

But with feminine la, there is no contraction:

• a + la papelera = a la papelera

So al papelera is incorrect.

The present tense here usually suggests a habitual action or a general sequence:

• After reading the magazine, my sister throws it in the bin.

So the sentence can mean this is what she normally does.

Spanish, like English, can use the present tense to describe a routine:

• lee la revista
• la tira a la papelera

The fact that one action happens before the other is already shown by después de.

In Spain, papelera commonly means a wastepaper bin, trash bin, or small bin, especially one indoors or in a public place.

So here it means something like:

• bin
• wastebasket

Depending on the country, other words are also common, such as:

• basura
• cubo de basura
• bote de basura

But papelera is very natural in Spain.

Because Después de leer la revista is an introductory time phrase, and Spanish often separates that kind of opening phrase with a comma:

• Después de leer la revista, mi hermana la tira a la papelera.

It helps mark the pause and makes the sentence easier to read. In short sentences, punctuation can vary a little, but the comma here is standard and natural.

Yes. That word order is also grammatical.

Both versions mean basically the same thing:

• Después de leer la revista, mi hermana la tira a la papelera.
• Mi hermana la tira a la papelera después de leer la revista.

The version with Después de leer la revista first gives the time/background information up front. That is why it feels slightly more natural if the speaker wants to emphasize the sequence of actions.