I only have the Kindle app for the Mac, but have you done whatever the equivalent is of 'Tools -> Sync and Check for New Items'?
For deleting, here is where my Maccy skillz suffer as I just want to right click on the book. But it seems like you select the book then go 'File -> Remove from Device'. It also seems like there's some kind of archiving option but I can't work out how that works.
Well I went on the sync thing on the kindle and it said there wasn't anything. I don't know it's like they are both 'linked' with my amazon account but not with each other (if indeed they are supposed to be?)
The weird thing is something I downloaded on the iPhone months ago (even though I deleted the app in the meantime, as I was using iBooks, but I got the kindle app again yesterday) has appeared on the kindle - yet the other things haven't. Maybe it's because they are only samples?
I could try going back on the kindle store and 'buying' them again for the kindle I guess, but obviously I wouldn't do that for a whole book as I might get charged twice...
If you have an iPhone I personally would recommend using iBooks instead of the kindle app. It's much easier to use, nicer to read and you can also download .epub files and drag them onto your iphone so you don't have to pay for them.