Hi Carissa,
Welcome to the forum,
I have never seen or heard anyone who would use ‘in behalf” in any context.
The expression is always “on behalf”, meaning representing.
On behalf of my family, I would like to take this time to thank everyone for coming.
As to the auxiliary words usage, it’s often depending on the mood and intent of the writer to express the willingness and certainty.
I will come to her wedding. Very certain
I would come if I have time- it’s conditional but possible.
I won’t go to her weeding- definitely not, a negative certainty.
I shall (rarely used) perhaps, I needed to …iffy
I must go to her wedding- very certain, a strong will
I might go to her wedding. Not very sure.
I hope this helps.