Your health insurance will go until the last day of the month of the month that you leave. If you leave any day in July, it will go until July 31. Keep this in mind with your new employer.
How long they keep you around after your announcement depends on your department and boss. Since you do not know how they will react, clear out as much of your personal belongings as you can before making the announcement. You may have to come in early or stay late so it isn't obvious.
Do not accept a counter offer. 80% of employees that accept a counter offer are gone in 6 months. Look up "5 reasons NOT to accept a counter offer".