I am wondering... Are demonstrations against the regime or against the president allowed?
Are demonstration against any regimes in any country in the world "allowed"?
What did the French regime do about 2 years ago when people were protesting the police brutality after they shot a 17-year-old kid in cold blood? They announced Martial Law!
And this is what they've been doing to the Yellow Vest movement in Europe for years:
What are they doing in US and Europe these days to students who are protesting their regimes' support of genocide in Palestine? They are beating them and arresting them and these are peaceful protestors who aren't even looking for a domestic regime change!
...
How big would you say a demonstration in favour of a laic (or simply without the Ayatolah) republic would be?
2 years ago there were riots in Iran, majority of whom were singing a "regime change tune" (or rather a
color revolution). At most a couple of hundred participated in it (out of 85 million population of Iran).
In the same month there was anniversary of the Islamic Revolution victory that toppled the US backed dictatorship. Tens of millions participated in that, a record high which was exactly because of these riots were threatening Iran's safety and were being globally advertised as "the majority", the advertisement which is exactly the reason why you are asking this question here.
How big is such a demonstration you ask? Not even a tiny one according to what we've seen.
Anyways, during those protests, there was quite brutal prosecution by the regime against protestors and I even recall there were some people who participated in the protests and one they got captured, they got a trial (perhaps not the fairest one) and got sentenced to death. Not sure on what the charges were about, but they were probably religious, something related to apostacy or rebellion against the state.
Nobody is arrested in Iran for "protesting". Any arrest is either for riots (damaging public or private property), acts of terrorism, acts of espionage and stuff of that nature.
As Trump's National Security Adviser John Bolton confessed on BBC in his live interview, those riots 2 years ago in Iran were orchestrated by United States (operation with the codename Zhina) and it US was arming foreign terrorists (mainly Kurdish separatists that were members of an international terrorist organization called Komoleh) and sending them into Iran.
What Bolton confessed to are acts of terrorism and espionage at the same time.
As for the death sentence, capital crimes such as first degree murder get that sentence. In all cases I've seen from 2 years ago there have been a public court and the videos of the murderers committing the crime (ie. indisputable evidence of the capital crimes) have been available. Such videos are available on the internet already, they're in Farsi though.
For example in one case that took place on Karaj-Tehran highway, I was close to the scene myself. A small group of terrorists shut down the highway by dumping large stones in the middle of it, creating traffic. Then they started attacking the stopped vehicles. In the middle of that chaos a young kid tries to clear a small path so that vehicles can escape. The terrorists seeing this start attacking him with knives and stones (and later they bring out their guns) and brutally murder the kid. Here is some screenshots of the video they recorded themselves of their acts of murder!
The photo on the top shows the murderer hitting the already dying kid in the face with his own shoe.
The photo on the bottom right shows the murderer pulling the dead body of the kid to the side, in front of the cars to block the cleared path and continue the blockade.
The trial these murderers received wasn't just fair it was also more than they deserved.