Check play state of AVPlayer

You can tell it's playing using:

AVPlayer *player = ...
if ((player.rate != 0) && (player.error == nil)) {
    // player is playing
}

Swift 3 extension:

extension AVPlayer {
    var isPlaying: Bool {
        return rate != 0 && error == nil
    }
}

In iOS10, there's a built in property for this now: timeControlStatus

For example, this function plays or pauses the avPlayer based on it's status and updates the play/pause button appropriately.

@IBAction func btnPlayPauseTap(_ sender: Any) {
    if aPlayer.timeControlStatus == .playing {
        aPlayer.pause()
        btnPlay.setImage(UIImage(named: "control-play"), for: .normal)
    } else if aPlayer.timeControlStatus == .paused {
        aPlayer.play()
        btnPlay.setImage(UIImage(named: "control-pause"), for: .normal)
    }
}

As for your second question, to know if the avPlayer reached the end, the easiest thing to do would be to set up a notification.

NotificationCenter.default.addObserver(self, selector: #selector(self.didPlayToEnd), name: .AVPlayerItemDidPlayToEndTime, object: nil)

When it gets to the end, for example, you can have it rewind to the beginning of the video and reset the Pause button to Play.

@objc func didPlayToEnd() {
    aPlayer.seek(to: CMTimeMakeWithSeconds(0, 1))
    btnPlay.setImage(UIImage(named: "control-play"), for: .normal)
}

These examples are useful if you're creating your own controls, but if you use a AVPlayerViewController, then the controls come built in.