the preposition befor DAY
 

.....

Re: the preposition befor DAY
 

Here the preposition has nothing to do with the noun day, but is dependent on participation :

You need in with all expressions related to the verb "participate" .

He was unwilling to participate in the demonstration.
Are you going to take part in the protest.
I expect your active participation in the lesson.

In your example you could change the end and the in would remain :

Union leaders called for the active participation of all members in the strike

showing that it depends on the noun participation, not on what follows.

So, in other words, here the prepositional phrase in the day of protest is the complement of of the verb, and not a time adverbial - as would be "on the day of the protest". Compare :

We are organising a strike. All members are expected to participate in the strike.
We are organising a day of protest.All members are expected to participate in the day of protest.

We are organising a strike on Thursday. All members are expected to participate on Thursday.
We are organising a protest The protest is on Thursday. All members are expected to participate on the day of the protest.


We are

Re: the preposition befor DAY
 

......

Re: the preposition befor DAY
 

Why should it be "in" for arrive? Arrive, unlike participate, is not a prepositional verb and doesn't have an associated preposition. It's an ordinary non-transitive one word verb. Because of the meaning it will often be followed by a prepositional phrase : "arrive" implies a place and a moment in time, and prepositions express location and time. But which preposition is used is dependent on the following noun, not on "arrive" :

We arrived / at the station
We arrived / on time
We arrived / in London
We arrived / behind schedule
We arrived / on the island
We arrived / at ten o'clock
Fall has arrived /above the treeline
... the boats arrived / below the fort
etc etc

Or of course there may be no prepositional phrase at all :
We arrived early
We arrived and checked in

Re: the preposition befor DAY
 

......

Re: the preposition befor DAY
 

No. It's as I said - is the preposition dependent on the verb (and therefore must always be present or retrievable from context) or is it omissible ? For example "ask for" meaning request. Without the preposition, or with a different preposition the meaning of the verb changes completely :

He asked for a pencil. = requested
He asked about my mother = enquired

You can't use ask meaning request without for. But ask meaning enquire can continue in many ways, not just with a prepositional phrase - eg :
He asked if my mother was OK / He asked what time the train was due.

So ask for meaning request is a prepositional verb - a unit of meaning. Ask meaning enquire, on the other hand, is just a normal verb followed, by chance in some cases, by a prepositional phrase.

Re: the preposition befor DAY
 

No. It's as I said - is the preposition dependent on the verb (and therefore must always be present or retrievable from context) or is it omissible ? For example "ask for" meaning request. Without the preposition, or with a different preposition the meaning of the verb changes completely :

He asked for a pencil. = requested
He asked about my mother = enquired

You can't use ask meaning request without for. But ask meaning enquire can continue in many ways, not just with a prepositional phrase - eg :
He asked if my mother was OK / He asked what time the train was due.

So ask for meaning request is a prepositional verb - a unit of meaning. Ask meaning enquire, on the other hand, is just a normal verb followed, by chance in some cases, by a prepositional phrase.