Why would Zachariah doubt Angel Gabriel? My point is:
Were they used to seeing false angels such that he'd doubt one even in the place of the incense?   

As a priest, he'd probably know a lot of Jewish history and the numerous appearances of Angels from God
Zechariah had a number of precedents which would have been an example to him.
Firstly, Abraham had been visited by three 'men' and the narrative is written in a mysterious way, indicating the nature of that 'visitation' :

And he lift up his eyes and looked, and, lo, three men stood by him: and when he saw them, he ran to meet them from the tent door, and bowed himself toward the ground, And said, My Lord . . . .  [Genesis 18 2,3 KJV]
And they said unto him, Where is Sarah thy wife?  [verse 9]
And he said, I will certainly return unto thee according to the time of life; and, lo, Sarah thy wife shall have a son. [verse 10]

Manoah's wife was barren and an angel visited them and a promise was made and Manoah said :

Now let thy words come to pass.  [Judges 13:12 KJV]

Manoah believed the angel. And so did Abraham believe the visitation he had received :

And being not weak in faith, he considered not his own body now dead, when he was about an hundred years old, neither yet the deadness of Sara's womb: He staggered not at the promise of God through unbelief; but was strong in faith, giving glory to God; And being fully persuaded that, what he had promised, he was able also to perform. And therefore it was imputed to him for righteousness.
[Romans 4: 19-22, KJV ]

When Zechariah prevaricates and says :

Whereby shall I know this? for I am an old man, and my wife well stricken in years ?
[Luke 1:18 KJV] . . .

. . . . the angel Gabriel responds :

I am Gabriel, that stand in the presence of God; and am sent to speak unto thee, and to shew thee these glad tidings. And, behold, thou shalt be dumb, [Luke 1: 19b, KJV]

One imagines the mighty Archangel drawing himself up in a slightly frosty response, somewhat indignant at the unbelief of the Levite priest. 'I am Gabriel . . . '
And Gabriel uses just a tiny fraction of his immense powers to chasten the Levite with dumbness for the next nine months.
Later, Mary shows no such unbelief and accepts a far, far greater message, only asking the perfectly legitimate question (there being no precedent to the import of the angelic revelation) 'How shall it be, seeing I know not a man ?'
Not, how shall I know this (for she has already been made aware of the fact) but she asks 'how' as to the means of the staggering miracle with which she is being presented.
So, in summary 'Why would he doubt ?' : well, he should not doubt, for there was ample precedent and example for him to believe and follow.
So why did he doubt ?
Maybe because for four hundred years there had been no further scripture written after Malachi's prophecy. Only there had been the translation of Hebrew into Greek (LXX).
Maybe because of what was going on in the temple, if the things that Jesus corrected, physically, had already begun.
Maybe because of his age and the weariness of waiting for the redemption of Israel.
Maybe because of the activities of scribes and Pharisees and doctors of the law and the Herodians and the relatives of the High Priest : again, assuming that what we see in the days of Jesus' ministry had already become established in Zechariah's day.
Maybe because of discouragement.
Maybe because of age.
Maybe because of weariness.
(I speak as a man of sixty eight years myself.)
Maybe all of this.
And who, then, shall point the finger and condemn ?
Not I.
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There is no evidence in scripture that 'false angels' were seen, either regularly or at all. And I can see no reason, therefore, for that to be a factor in Zechariah's mind as to doubting what had been said to him.
Had that been the case, the narrative would surely have informed us of that factor.