I played and it sounds favorable and seems like each beat note are getting there count which they use to hold. Any way please make me confirm by telling how it's fine if I didn't made mistake and if I made mistake tell me. I would love to hear the reason no matter correct or wrong. I just need solid reason of this beat count which I am little unsure of it.
I'm guessing it's being strummed on guitar. Yes, the rhythm, is written properly - not sure why there's a little dot after the a of beat 3, 1st bar.
On guitar, it's easier to play ghost strums where there are no chords sounding. It keeps the rhythmic pattern of the strumming hand/arm going. Far better than just stopping, and starting again at the next strum.
To 'play' the rests, just release the pressure of the fretting fingers, in the same way as you do with the chords marked with 'x'.
There's nothing wrong with counting the rests in the same '2e&a' way. Just because it's a rest doesn't mean it's different from a note - both have counting values!