Both the slide and your picking hand should be playing an 8th note rhythm. So don’t worry about if you wait till you end the slide to play the open 2nd string. Focus on the rhythm of each hand. You need both hands to operate independently which is tricky.
Yes, you’ll definitely want to play the open string after the slide. That is how you keep the “roll” of your right hand going. If you didn’t play the open string with the slide, your right hand picking rhythm would break.
With bluegrass style slides like these, you want to keep the rhythm going with your right hand. The slide is basically extra credit.