This behaviour is changing in version 1.8 (now in beta), because you aren't the only person with this question. If only one track is selected, the whole list will transfer to the Now Playlist list, with the position set at the selected track.
For now, in 1.7, if you have any songs selected when you press play, only those songs will be played. So, you can either select the entire playlist, or select none of the playlist, and press play, to get the entire playlist. As a shortcut, just double-click on an album (or anything else), and it will start playing.
Album art is coming from either embedded images, or folder images. Typically if you are seeing wrong cover art you didn't expect, it means you don't have folder images set up like we expect - you can disable folder images from the preferences. (There's also the issue of whether you want playing artwork, or select artwork, as justice mentioned).
Prev/Next buttons only affect the position in the "Now Playing" list, depending on where you are currently in that list.
You can also use drag and drop to the Now Playing list to put things in the queue.
Hope that helps!