Can one search by number of miles given for each $?
 

There seemed to be way to search for those restaurants whivh give 10 miles? Any suggestion to find it?

For American Airlines, try

http://aa.rewardsnetwork.com/diningdeals.jsp

After logging in to AA IDine, there was a "Hot Deals" listing on the cover page,

Have you tried doing a dining search and clicking "Sort results by reward"?

I just learned that one could search for the special value deals where they show who still gives 10 miles per $ for the entire country.

Are you referring to restaurants that give 10/$ normally, plus the elite bonus of 10/$ - for a total of 20/$ - or the "regular" locations that give 5/$, plus the elite bonus of 5/$ - for a total of 10/$? I thought being elite, almost ALL restaurants gave you 10/$ - after the bonus? So wouldn't almost 100% of the locations be 10/$?

Concerning hot deals, here's the link to the UA rewards network site:

http://mpdining.rewardsnetwork.com/diningdeals.jsp

It shows the restaurants in every state that are offering the 20 miles/ $ deal for elite members. As people have noted in other threads, beware these restaurants. Many aren't of the highest quality. For example, in the Chicago area, only Maza is considered to be a good "Hot Deals" restaurant.