If there is something that represents disbelief on the door of a shop, like a picture or writing, and so on, is it an act of disbelief if I enter the shop and leave the shop by closing and opening the door? Another example: if there is something that represents disbelief on someone's clothing and they ask me to hand it over to them, is it an act of disbelief? I hope you could ensure to give me a very detailed answer as I might come across other such situations.

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.

What you mentioned are not acts of disbelief. The scholars of Fiqh allocated a chapter in their writings to apostasy and the actions that entail it, and what you mentioned in the question is not among them.

Takfeer (judging oneself or others to be disbelievers) is a serious matter. It is based firstly on establishing evidence that a certain action is an act of Kufr (disbelief), and secondly, that the conditions of Takfeer are met and its impediments are absent. Shaykh Ibn ‘Uthaymeen said:

"It is impermissible to say that so and so is a Kaafir (disbeliever) until two matters are verified: The first is to verify that what he said or did is an act of disbelief that takes the doer out of the fold of Islam. If there is doubt about the matter and it is not known with certainty, then it is impermissible to declare him a disbeliever, and the basic principle is that he remains Muslim... The second is to verify that the description of Kaafir applies to this person ... such that he meets the conditions of Takfeer. Among the conditions is that he knows (that what he did is Kufr) and intended it. If he did not know, then we do not accuse him of Kufr because evidence has not been established against him (i.e. he was not warned that what he did is Kufr." [Liqaa' Al-Baab Al-Maftooh] 

It seems that you are suffering from Waswaas (obsessive Satanic whispers) regarding Takfeer. If you wish to be safe from the evil of this disease, then ignore this Waswaas and seek refuge with Allah from the accursed Satan in addition to reciting Ruqyah (healing through Quranic recitation and supplications) over yourself.

For more benefit, please refer to fatwas 88660 and 82918.

Allah knows best.