Chciałbym lepiej poznać twoją siostrę.

A natural word-by-word breakdown is:

• Chciałbym = I would like (said by a male speaker)
• lepiej = better
• poznać = to get to know
• twoją = your
• siostrę = sister

So the whole sentence is literally something like:

I would like to get to know your sister better.

Because chciałbym sounds softer, more polite, and less direct.

• Chcę lepiej poznać twoją siostrę = I want to get to know your sister better
• Chciałbym lepiej poznać twoją siostrę = I’d like to get to know your sister better

In Polish, using chciałbym / chciałabym is very common when you want to sound tactful or polite.

Yes. Chciałbym tells you the speaker is male.

It is built from:

• chciał- = a masculine past-form stem
• -bym = a conditional element, roughly giving the sense of would

So:

• chciałbym = I would like (male speaker)
• chciałabym = I would like (female speaker)

This is very normal in Polish: some verb forms show the speaker’s gender.

Because siostra is the dictionary form, but here the word is the direct object of the verb poznać.

After poznać kogoś (to get to know someone), you need the accusative case.

So:

• siostra = nominative
• siostrę = accusative

Many feminine nouns ending in -a change to -ę in the accusative singular.

Because twoją has to match siostrę in gender, number, and case.

Since siostrę is:

• feminine
• singular
• accusative

the possessive adjective must also be:

• feminine
• singular
• accusative

So:

• twoja siostra = your sister (subject form / nominative)
• twoją siostrę = your sister (object form / accusative)

Because poznać means to get to know, while znać means to know already.

Compare:

• Znam twoją siostrę. = I know your sister.
• Chciałbym poznać twoją siostrę. = I’d like to get to know your sister.

So poznać is the right choice when the relationship or familiarity is still developing.

Poznać is perfective.

That fits well here because it expresses reaching a result: getting to know someone.

Its imperfective partner is poznawać.

• poznać = to get to know, to come to know
• poznawać = to be getting to know, to get to know over time / repeatedly

In this sentence, poznać is the normal and most natural choice.

A sentence with poznawać is possible in some contexts, but it would focus more on the ongoing process rather than the result.

Lepiej means better, and in this sentence it modifies poznać.

So lepiej poznać kogoś means:

to get to know someone better

This is a very common and natural expression in Polish.

It does not mean that you already know the person extremely well. It can simply mean you want to know them more than you do now.

No, Polish word order is fairly flexible, because the endings show the grammatical roles.

The version you have is the most neutral and natural:

Chciałbym lepiej poznać twoją siostrę.

But other orders are possible for emphasis, for example:

• Twoją siostrę chciałbym lepiej poznać.
  This puts extra focus on your sister.

Even when the word order changes, the case endings still show who is doing what.

Yes. Twoją comes from twój, so it is used when speaking to someone with ty.

If you want to be formal, Polish usually uses Pani or Pana instead:

• Chciałbym lepiej poznać Pani siostrę. = speaking formally to a woman
• Chciałbym lepiej poznać Pana siostrę. = speaking formally to a man

So the original sentence is informal or familiar.

Not if you mean your sister.

• twoją siostrę = your sister
• swoją siostrę = one’s own sister / his own / her own / my own, depending on the subject

So:

• Chciałbym lepiej poznać twoją siostrę = I’d like to get to know your sister better
• Chciałbym lepiej poznać swoją siostrę = I’d like to get to know my own sister better

That is a big meaning change, so they are not interchangeable.