A PBS show called "Odd Squad" had open auditions. Approximately how many children audition for each role?

No way to know, but I would guess thousands of self-tape auditions will be sent in and at least half of them will be ignored because they’re not appropriate.  The announcement I saw specifically says the talent must reside in Canada but many kids and parents will ignore that or they’ll not follow the explicit submission directions especially if they are unfamiliar with how the industry works. 

PBS is looking for kids 7-10 years old so with kids that young CDs will want to cast a wide net.  Any good agent or manager will work to contact the CD directly to pitch the kids they represent to make sure their videos gets seen. (The application even mentions that if the child has an agent to let the agent submit the child’s audition package.)  I suspect that most other tapes won’t be watched past the slate.

But even the experience of making a self tape can be fun and be a learning experience and there’s no harm in trying as long as everyone is very realistic about their expectations. 

That's impossible to answer.

But typically, a production company would find thousands of children lined up outside - even tens of thousands.  They'd then have their casting people walk up and down briefly looking at the children and picking out any who have the right physical appearance for each character they're casting.  Those would be asked to go to a different area where they'd be asked if they have had any real training and any experience. The ones who have MIGHT be asked to read a few lines to camera, leaving their name and contact details.  If any of those are talented and have what they're looking for, they might be considered for a role.But the reality is that almost all 'open' auditions are nothing more than a publicity stunt.  Usually, they've already cast the roles.

NOBODY HERE IS GOING TO KNOW THE ANSWER. iF SOMEBODY CLAIMS THEY DO THEY'RE TROLLING YOU.