Le agradezco a mi suegra por sus consejos.

Le is an indirect object pronoun meaning to him / to her / to you (formal). In this sentence it stands for the person receiving the thanks—your mother-in-law.
So Le agradezco... = I thank her... (literally: I thank to her...).

This is a very common Spanish pattern called indirect object doubling: you use the pronoun (le) and you can also name the person (a mi suegra) for clarity or emphasis.

• Le agradezco a mi suegra... (clear, natural)
• Le agradezco... (if it’s already obvious who)
• A mi suegra le agradezco... (more emphasis: It’s my mother-in-law I’m thanking)

In Latin American Spanish, this doubling is especially normal.

Because agradecer takes an indirect object (the person you thank), and indirect objects are commonly introduced with a.
So a mi suegra = to my mother-in-law (even though English usually just says I thank my mother-in-law).

Yes, it’s grammatically possible: Agradezco a mi suegra por sus consejos.
But many native speakers still prefer Le agradezco a mi suegra... because the pronoun makes the structure feel more complete and conversational.

They focus on different things:

• Le agradezco a mi suegra por sus consejos. = I’m thanking her for the advice (the person is central).
• Agradezco los consejos de mi suegra. = I appreciate/thank the advice from my mother-in-law (the thing is central; it can sound a bit more formal/written).

With agradecer, the reason you’re thankful is introduced with por:

• Gracias por... / Le agradezco por... = thanks for...

Para is more like for (a purpose) / in order to:

• Esto es para ti. = This is for you (intended recipient).

So here por is the natural choice.

Sus can mean his/her/their/your (formal). It’s ambiguous by itself, so you rely on context. Here, it normally means her (your mother-in-law’s) advice: her advice.

If you need to be extra clear, you can say:

• ...por los consejos de ella (her advice)
• ...por sus consejos is still the most natural in most contexts.

Plural consejos is common because it suggests she has given you multiple pieces of advice over time. Singular is also possible:

• ...por su consejo. = for her advice (as a general idea) or for one specific piece of advice.

Both are correct; the nuance depends on what you mean.

Agradezco is present tense, first person singular of agradecer: I thank / I am grateful.

It can express:

• a general habit (I’m grateful to her)
• a current feeling (I’m thanking her right now)
  Context determines which is intended.

If you’re speaking to your mother-in-law:

• Usted: Le agradezco (a usted) por sus consejos.
• Tú: Te agradezco por tus consejos.

Notice the changes:

• le → te
• sus → tus (because the advice belongs to the person you’re addressing)

Yes. Agradecerle is just the infinitive agradecer + le attached: to thank him/her/you(formal).
Examples:

• Quiero agradecerle a mi suegra por sus consejos. = I want to thank my mother-in-law for her advice.
• Le agradezco a mi suegra... = I thank my mother-in-law...

Same meaning; different grammar (infinitive vs conjugated verb).

It’s neutral-to-formal mainly because agradecer is a bit more formal than simply saying gracias. In casual speech you might also hear:

• Gracias a mi suegra por sus consejos.
• Gracias por los consejos, suegra. (if you address her directly)

But your original sentence is perfectly natural and polite.