Most (if not all) webhosts have their own DNS management (usually hosted on a number of servers) but they are 2 separate services (although often sold/supplied together).

So you will still need your domain seller to provide DNS to point to your server. If you don't have a domain then TKL Hub can provide DNS (and free domains).

Most systems do not run their own DNS server, but utilize their /etc/hosts file for name resolution and then rely upon external DNS servers for things not in /etc/hosts.

so, if there is a hostname you need to resolve on your lamp server and it's not in DNS, just add a line to /etc/hosts and you'll be good.

Note, this won't help other systems resolve the address. For that, yes... you should setup DNS resolution and then add the DNS server to all client machines. 

There's nothing preventing you from adding a DNS server to the LAMP stack. I haven't used bind in a LONG time some wouldn't be the best person to answer. I'd suggest looking at tutorials for installing bind on a Debian system, as TKL is based on Debian. The steps should be identical.

After you add the DNS server to your lamp stack, it won't mean that clients will automatically be able to resolve the names unless you add the DNS server to the clients resolution order. I think, and could easily be wrong, that your DNS must be 'authoratative' for the domains it serves in order to propagate to other DNS servers.

Again, I'm not a bind or DNS person but would suggest looking for tutorials on Debian and Bind... that should point you in the right direction.