Is it predestined how many children we will have? If someone don`t take any birth control system but make dua for not having children for 6 years does that mean without tying the camel putting trust in Allah(swt)? Women put something inside their vaginal area for not having kids for 3 or 5 years, are these 2 methods permissible?

All praise be to Allah, and may His blessings and peace be on the final messenger, Muhammad,

Everything is predestined, but we should always take precautions. There is no conflict between these two assertions. The Prophet (blessings and peace be upon him) said,

"اعملوا، فكل ميسر لما خلق له".

"Do perform good deeds, for everyone is facilitated in that for which he has been created”

Adopting all causes and means is an obligation, and believing in the means as ultimate effectors is polytheism: The Messenger (blessings and peace be upon him) said,

"احرص على ما ينفعك واستعن بالله".

“Watch over what is good for you, and seek refuge in Allah"

As for birth control, the default principle is that fertility is a blessing and Muslims should aim to have large families. This is due to the saying of Allah’s Messenger (Allah's peace and blessings be upon him):

"تزوجوا توالدوا تناسلوا فإني مباه بكم الأمم يوم القيامة."

   “Marry and re-produce, for I will be proud of you on the Day of Resurrection.”

It is a given that this is recommended within man’s capacity. If there is a reason for birth control, such as the mother’s weakness or her inability to look after the various needs of more kids, then it is allowable to have some planning. Jabir ibn Abdillah said:

"كنا نعزل والقرآن ينزل."

“We used to practice coitus interruptus while the Quran was descending.”

Birth control is not obligatory, but only permissible, given there is no greater harm involved. Therefore, if someone decided to make du’a’ instead of using medicine or any birth control methods, they may do that. You should take precautions if you truly want to avoid conception. You may consult your physician regarding the various methods of contraception, which include non-pharmacological methods.

Using IUDs is another contraceptive method. There are other methods as well, and you should consult your physician regarding the balance between the harm and benefit of those methods. If that method is not harmful, then it should be permissible. To my knowledge, this method is one of the safer and more effective methods.

Allah knows best.