Hello vmorton,
I just checked our tickets from Venice last year and I paid full price at St Marks but did have a senior discount at Dodge's Palace. This was for the full Secret tour not just a ticket, but the website uses the same language so for the senior discount I would just go ahead and buy it.
Even the places that warned "we will check IDs at time of entry" on the website for senior pricing never seemed to worry about it when we actually arrived. If you had a ticket you had a ticket. But I can't say for sure the same is true for the student discounts.
Many smaller museums offer senior discounts regardless of nationality versus the bigger sites - some were even free. At the Ferragamo museum in Florence - the woman at the desk very discretely mentioned 'there may be discounts that may benefit you if you are interested' in the classiest manner possible while we were buying tickets.
Hope that helps,
=Tod
PS The museum upgrade at the Basilica is very much worth it.