Can Lingual Braces Correct Crossbite and Also with Diastema (Upper)?

Lingual braces work similar to the outside braces so they should be able to correct the same things. Whether your cross bite can be corrected will depend more on your age and if there is an underlining jaw problem causing it and not as much on what kind of braces you use.

A crossbite is when the width of the upper arch is narrower than the bottom arch.  An expander is a device that widens the upper arch and is generally used when the crossbite in the upper arch is moderate to severe.  Orthodontics alone whether it is invisalign or brackets with a wire are pretty good correcting mild crossbites however may not be successful in correcting severe crossbites.  So depending on your crossbite you may need to add an expander to get the desired result.

Both of the issues you described can be addressed with lingual braces.  However, depending on the type of cross bite and your age, it may be challenging with either normal braces or linguals.  I definitely recommend a custom lingual bracket system such as Incognito.  Faster, more predictable results.  A consultation with an orthodontic specialist, who does linguals (there are only a few) will help you answer the question, based on your particular bite.