I know that using LocalDb is very good and easy for developement, I wonder if it's good idea to use it in production when I host websites on IIS server?
I'm asking because I wonder if it won't have any kind of speed issues in production.
This is my connection string that I want to use in production
Server=(LocalDB)\\v11.0;Integrated Security=SSPI;MultipleActiveResultSets=true;
AttachDBFilename=|DataDirectory|ProjectDB.mdf;
LocalDB is absolutely supported in production. From the performance point of view it is identical to SQL Server Express, as they share the same database engine.
Now, for a site running in full IIS I would recommend using service-based instance of SQL Server Express. Since IIS is running as a service it is best to have a service-hosted database as well. That means you should not be using User Instance=true or AttachDbFileName=... in your connection string.
If you want to learn more, I recommend this post on using LocalDB from full IIS, part 1 and its follow up, part 2. They go into more details.
"Normal" - hard to say, that's a relative term.
Permitted - yes. Here is a quote from the Microsoft SQL Server Express blog where LocalDB was introduced:
"...if the simplicity (and limitations) of LocalDB fit the needs of the target application environment, developers can continue using it in production, as LocalDB makes a pretty good embedded database too."
UPDATE (March 2019)
After using LocalDB in dozens of production sites for two solid years now, I'll confirm that it works. However, if you need a free multi-user ("shared" in LocalDB terminology) DB, use SQL Server Express instead. LocalDB is not a good multi-user solution. There are MANY obstacles to overcome. I don't have time now, but if you would like details on those obstacles and respective workarounds, leave a comment, and I'll document them here or in another question.
I don't think so,
Even though LocalDb and SQLExpress might be the same on performance, LocalDb's Process shuts down itself if it doesn't have any request for a while. And usually the next request takes too long to response (or even worse, it times out).
I would recommend you install your own instance of SQLEXPRESS.
