Can I replicate/connect/whatever to SQL Server over the Internet?

A. The Internet is nothing more or less than a large tcp-ip wan (wide-area network). And a wide-area network works the same as a lan (local-area network) just slower. So yes, anything you can do with SQL Server on the network in your office you can do over the internet, assuming you HAVE CONNECTIVITY to the internet. All issues are networking related not SQL related. The servers at either end must be able to connect to the internet either directly or via RAS. In the case of RAS you can use RASDIAL.EXE to automate connections as necessary.

Issues are :-

1. Internet is tcp-ip only, so you must run tcp-ip on your clients and the server(s)
2. Physical connectivity. Do you have a routed/dial-up connection to the internet? Are these routes being passed transparently into your internal network? You need to be able to PING and PING from your local client/server and vice-versa.
3. Security. You should really use a firewall for any internet connection. Is this configured to allow your SQL traffic through? See firewall.txt in the FAQ for more info on port numbers for the various net-libs.
4. Name-resolution. You can address the remote server(s) by IP address directly, but the better method is to put an entry in your local hosts file (SYSTEM32DRIVERSETC) or DNS/WINS server. Then put this name in your connection string.