I wear the amulet all the time. It is supposed to stack with the Blessing. Also, there is another item in the game that reduces the time between shouts 20%. So, it is possible to have 60% (or so) less time between shouts.
The Thalmor have a hit squad like other factions do and if you do certain things in the MQ, they come after you. Otherwise, you have to have a discussion about religion with them to get them to play.
I've had an awful lot of difficulty trying to get the Thalmor to come after me...I openly mug them when I see them (hey, the fine is less than a parking ticket for my horse, and the booty earns you a good living), I'm rude, I made no secret of my dislike for them at any opportunity, and hey, I even call in at the embassy and give them a stormcall whenever I'm in the neighbourhood...and after 600 hours of playing like this I've only recently had a Thalmor tell me I'm on their hit list...and never once have I come across any of them specifically coming after me.
I don't know what I'm doing wrong...it's very frustrating.