Question 1 :
Does modern Portuguese (both in Portugal and Brazil) still use article before demonstrative adjective or not?
For examples :
o meu casa / minha casa
a tua filha / tua filha
o nosso livro / nosso livro
a vossa camisa / vossa camisa

Confusing demonstrative adjectives 👈
seu, sua, seus, suas have ambiguous meaning, (his, her, your, their). To avoid confusion, Portuguese tends to use dele (his), dela (her), deles (their), delas (their).

Question 2 :
Does modern Portuguese, (both in Portugal and Brazil) use "seu, sua, seus, suas" as "you" or still use them as "his, her, your and their"?

Indirect objective pronouns 👈
lhe (singular) means his, her, you, them
lhes (plural) means his, her, you, them

Question 3 :
Does modern Portuguese (both in Portugal and Brazil) still use "lhe, and lhes", how to do overcome this confusion?

Prepositional pronouns 👈
"si" means him, her, you, them, but si can be replaced with "ele, ela, você, eles, elas and vocês".
For examples :
de si / dele/dela/de você/deles/delas/de vocês
para si / para ele/ela/você/eles/elas/vocês
por si - por ele/ela/você/eles/elas/vocês

Question 4 :
Does modern Portuguese (both in Portugal and Brazil) still use "si"?

Question

Question 1 : 
Does modern Portuguese (both in Portugal and Brazil) still use article before demonstrative adjective or not?
For examples :
o meu casa / minha casa
a tua filha  / tua filha 
o nosso livro / nosso livro 
a vossa camisa  / vossa camisa

Confusing demonstrative adjectives 👈
seu, sua, seus, suas have ambiguous meaning, (his, her, your, their). To avoid confusion, Portuguese tends to use dele (his), dela (her), deles (their), delas (their).

Question 2 :
Does modern Portuguese, (both in Portugal and Brazil) use "seu, sua, seus, suas" as "you" or still use them as "his, her, your and their"?

Indirect objective pronouns 👈
lhe (singular) means his, her, you, them
lhes (plural) means his, her, you, them

Question 3 :
Does modern Portuguese (both in Portugal and Brazil) still use "lhe, and lhes", how to do overcome this confusion?

Prepositional pronouns 👈
"si" means him, her, you, them, but si can be replaced with "ele, ela, você, eles, elas and vocês". 
For examples : 
de si / dele/dela/de você/deles/delas/de vocês
para si / para ele/ela/você/eles/elas/vocês
por si - por ele/ela/você/eles/elas/vocês

Question 4 :
Does modern Portuguese (both in Portugal and Brazil) still use "si"?

Answer

3: do you mean pronomes átomos?? Like dizer-lhe? Dar-lhe? Dar-vos? Etc

In European portuguese we use yes

Answer

1: in European Portuguese we always use the articles

Answer

What do you mean with modern Portuguese? That doesnt exist

Answer

Hello Andreu, 
I'll be answering your questions based on Brazilian Portuguese.

Question 1: Most of the time we use the article but it's ok if you don't. Both forms are acceptable.

Question 2: Yes, we do but not in spoken language, we prefer to use "dele, dela". You might find this in writing though.
When using "dele/a" pay attention to the word order and gender agreement: 
a sua casa/a casa dele/dela; o seu carro/o carro dele/dela 
seu/sua - the gender agreement is always with the noun.

Question 3: "lhe" is used predominantly in written language. I can see why you find it confusing. Most people avoid this one even in writing. I'll leave some examples to help you:
"Lhe":  a você, a ele, a ela,
- Eu dei dinheiro para ele/ela (spoken) // Eu lhe dei dinheiro (written).
- Eu não devo nada para você/Eu não te devo nada(spoken) // Eu não lhe devo nada (written).

Question 4: "si" is only used in very formal written language, or when the meaning is ‘oneself’. See the sentences below, they are more common:
Ele prometeu a ele mesmo que não voltaria a fumar.
Ela se surpreendeu com ela mesma.
Useful phrase with "si": por si só (by itself)

I hope I could help you.

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