You can, of course, ADD the lang value to the query string (from the path to search) but the easiest way would be to simply make it a hidden field in your form where it would be appended automatically.
As for your mod_rewrite, %2& means NOTHING in a URI (your first RewriteRule set means nothing … to me). Is that just a typo (for %24 which would be assumed to be $, a reserved character, in front of lang)?
Finally, I am very pedantic about abusing trailing slashes as they are intended to signify that the preceding “filename” is a directory rather than a file. Worse, it offsets internal requests for supporting files by one directory level which generally result in 404’s.