Well, the simplest way is to use Gmail as your primary mail provider. It's easy to do, it will go to your other mail carrier and suck in the mail for you whenever mail arrives.
I don't know how to select all in Gmail on a phone. You'll have to do that on a web based browser on a PC. It may work on a phone based browser but I haven't tried it.
On my old Droid Incredible, you just hit delete, then delete all and you were done. There has to be a way to do this from the ICS email app. I'm calling Verizon and will report back.
For SMS conversations you can long press on a message then select all the rest of the conversations you want to delete. It would be nice to see that in the email.
Maybe they don't want people to accidentally delete 200 emails cause it is deceiving you only see 10 but select all delete would represent like 200 emails.
Tick the box next to any of the emails, then click on the menu button (ON mine it is the three dots in a stack) and it will give you a select all and delete all options.
It will delete EVERY email in there so click carefully