How can I pass a value from block to page in a view?
I think this question may have some similarities to How to pass a value from form(block) to the page and show this page?

The trouble with what you are asking is trying to have two distinct parts of your page act on one interaction: the block form submit and then the page search view filter form submit. Each is in fact a separate function which have no relationship in the Drupal backend. To get this to work you could use some jQuery magic to get the value selected in Block filter be cloned into the page search and do a submit on the form. You may have to do some coding using cookies or session values to ensure the values are also reloaded into the block filter criteria if you wish to keep those synchronized. 

Since you have several ways of going about this, stick to one that is easiest to work with at first. jQuery is often the most convenient though maybe less elegant in some cases.