If I make a run on a remote server, (rezzed or unrezzed) and it has a piece of ice (rezzed or unrezzed) and I gain access and I can trash the card, after it trashes does the ice that protected it go away? Can he put a new asset or agenda in the same spot with the ice still protecting it? So can ice be there and NOT protect anything?

Yes, the server persists as long as there is some card in the server or protecting the server. If there is no ice and the installed card(s) is trashed, then it ceases to exist. It's in the main rulebook on page 6: "Ice can protect an empty server." And then on page 13, "Agendas, assets, upgrades, and ice can all be used to create a new remote server. If the Corporation creates a remote server by installing ice, the server exists, but is considered to be empty. An empty server can still be run against by the Runner." It's not explicitly stated--because it would be an infinite undertaking to list all the things that don't occur at various points--but ice is not trashed when the asset/upgrade/agenda they're protecting gets uninstalled for whatever reason. (There is no rule that says to do so, so you don't.)

Is there any way a Runner can trash an existing piece of Ice? Rezzed or unrezzed?

Not without specific card effects. Installed ice is never "accessed" so you can't pay any trash cost. Almost all ice doesn't even have a trash cost.

Card effects which can trash installed ice: Run Amok, En Passant, Apocalypse, Forged Activation Orders, Forked/Spooned/Knifed, Immolation Script, Kraken, Prey, and far and away the most commonly played, Parasite.

Thanks! As a Runner I don't like Ice.

Generally no.

Thanks, this tip helps. So, just to be clear, the corporation can create a new service by playing ice, and then at a later time, instal an agenda or asset in that server, behind the ice? Likewise, if a card behind the ice gets trashed/stolen, the ice does not go away and the corporate player can install a new agenda (or whatever) behind the ice, in the old space?

Thanks!

Precisely.

But if there is no ice, and the cards which were installed unprotected get trashed or otherwise uninstalled that server goes *poof* and ceases to exist.

I was playing a game yesterday and the runner installed a Temujin Contract and chose an unprotected server, which turned out to be nothing more than a 15 Minutes.