Play all songs in iTunes

The only way you'll get iTunes to play all the songs sequentially in a playlist or library is to have them checked, it's a design feature of the check boxes. Using unchecking to manage an iPhone or an iPod is an option (and works fine when you are working with a small number of specialised tracks such as Christmas songs that you want to keep off the iPhone or iPod during the year for example) but only if you can live with the other things unchecking does:
Unchecked songs will be skipped when playing songs from the library or a selection. They will only play when individually clicked on.
Unchecking is not selective, an item is either checked or it's not so it affects all the playlists an item might appear in.
Smart Playlists have an option that unchecked songs won't appear.
Unchecked songs will not be burned to CD.
Genius and iTunesDJ will not use the songs that are unchecked.

Basically unchecking is a way of telling iTunes to ignore the item and exclude it from other automatic features.

To avoid the problems of unchecking, another method you can use to auto-sync your iPhone with iTunes is "Sync Music - Selected Playlists". Syncing selected playlists is just like syncing your whole library only on a smaller scale and more "selectively". This is particularly useful when your library is larger than the capacity of your iPhone. When your iPhone is is set to sync selected playlists it automatically picks up any changes you made to that playlist in iTunes since you last connected. Anything you added to the playlist will be added to the iPhone, anything you deleted from it will be deleted from the iPhone. In other words the iTunes playlist and the iPhone are kept "in sync".

Open iTunes and create a playlist to update your iPhone from, call it -My iPhone for example (the dash will position it at the top of the playlists in Sources). Drag whatever content you want to have into the playlist. Now connect your iPhone and when it appears in the Source list click on the iPhone icon to bring up the preference tabs in the main pane. Go to the Music tab and choose Sync Music and the Selected playlists radio button. Choose the playlist you just made from the selection and click Apply. You can also sync from any existing playlists by choosing the same setting, you just need to make sure that the size of the playlists don't exceed the capacity of your iPhone.