why Threads_connected falls down automatically in some cases
It sounds like there are a few long queries that lock lots of things, then runs for a while.  Meanwhile, other connections come and try to get at the locked things, but get stalled.

If you are using MyISAM, switch to InnoDB.

Set long_query_time to 1 and turn on the slowlog.  After a while, it will catch the naughty query.  Show us the query, plus SHOW CREATE TABLE for the table(s) that it touches.