Is it blasphemy to question Islam?
No, it is not blasphemy to question your belief in Islam. The companions of the Prophet (pbuh) often asked him questions regarding Islam. During the time of the prophet, people often came to the prophet (pbuh) to ask him questions about Islam.
Narrated Abdullah bin Ammar: I saw the Prophet (ﷺ) near the Jamra
and the people were asking him questions (about religious problems)...
Narrated Ibn Abu Mulaika: Whenever `Aisha (the wife of the Prophet)
heard anything which she did not understand, she used to ask again
till she understood it completely...
Narrated Abu Musa: A man came to the Prophet (ﷺ) and asked, "O Allah's
Messenger (ﷺ)! What kind of fighting is in Allah's cause? ...
Narrated Abu Sa`id Al-Khudri: Some women requested the Prophet (ﷺ) to
fix a day for them as the men were taking all his time. On that he
promised them one day for religious lessons and commandments...
Narrated Abdullah bin Amr bin Al `Aas: Allah's Messenger (ﷺ) stopped
(for a while near the Jimar) at Mina during his last Hajj for the
people and they were asking him questions...
Is there any verse of Qur'an or hadith that talks about a certain degree of asking or a point that we should not exceed?
There is a verse which says "O you who believe! Ask not about things which, if made plain to you, may cause you trouble". Allah almighty, may he be gorified and exalted, says in the Qur'an (interpretation of the meaning) :
[5:101] “O you who believe! Ask not about things which, if made plain to you,
may cause you trouble. But if you ask about them while the Quran is
being revealed, they will be made plain to you. Allah has forgiven
that, and Allah is Oft-Forgiving, Most Forbearing”
But this prohibition applies specifically to the time of the Prophet (while the Quran was being revealed). This verse forbade the companions to ask the Prophet (peace be upon him) about matters that Allah had not said anything in the Qur'an. It also forbade them to ask about things that they didn't know of and if they came to know of them it could upset them
(e.g questions like, "who is my father?", "what did my father think of me"). This prohibition doesn't apply anymore.
At what moment our questions become blasphemous?
There is no evidence in the Qur'an or Sunnah that limits what we can ask. So, there is no "moment" when our questions become blasphemous. Asking questions is encouraged Islam. Allah says in the Qur'an (interpretation of the meaning) :
[16:43] And We sent not before you except men to whom We revealed [Our
message]. So ask the people of the message if you do not know.
Narrated Anas bin Malik: One day Allah's Messenger (ﷺ) came out
(before the people) and Abdullah bin Hudhafa stood up and asked (him)
"Who is my father?" The Prophet (ﷺ) replied, "Your father is Hudhafa."
The Prophet (ﷺ) told them repeatedly (in anger) to ask him anything
they liked. `Umar knelt down before the Prophet (ﷺ) and said thrice,
"We accept Allah as (our) Lord and Islam as (our) religion and
Muhammad as (our) Prophet." After that the Prophet (ﷺ) became silent.