Can bitcoind be installed and run on shared hosting? Has anyone successfully done this? If not why not.
A full instance of bitcoind or the client isn't needed.
You probably are looking for a "checkout button" to a server that will either host your wallet for you or will cash out on demand.
There are many, but MtGox has one and I'm pretty sure Bitpay has one as well. This isn't a comprehensive list, but the concept is that you add a "button" to your website that facilities payment. This button works with ASP.NET, PHP and anything else that generates HTML.
Note that although it may be possible for static HTML pages to be supported, I haven't investigated if its possible. (You haven't indicated if you're using plain old HTML or a dynamic language)
The answer is yes. But you keep an application open running it ,as you would a php torrent tracker. But the requirements won't be met for the depencies required.
So that being said.
Means yes ,but because of lacked dependency issues , that's the only thing stopping you. You need small cheap vps. Download a little SH file edit it and put proper depency files tonbe installef ,and you should be good to go.
Sidenote: Many people will not help you with such a thing. They will say off the wall things out if ingnorance or jealousy. BUT remember running such a thing like that deamon ,or mining from shared hosting can get your account nailed with no return of money. Because it's a violation of shared hosting terms.