I just started running a dedicated server with the help of another server host and at first things were fine.

However, I have two problems:
1. IP keeps changing.

2. If no map is chosen (Ex: Server Empty) the map sorta changes, but nothing shows up and it stays at the original map. Also, if I don't restart the server, you can't use attacks in the game.

Anyone have a solution? Particularly for the first one?

1. If the public IP keeps changing it's probably because your ISP assigns your address dynamically. Dynamically assigned addresses can happen whenever you sign on or at the end of a IP address "lease", sometimes both. There is not much you can do to stop this from happening other then requesting that they assign you a static IP. Most ISP's charge a small fee for assigning you a permanant IP address, so be prepared for the change in billing.

1a. If you mean your private IP keeps changing, then set it statically on the server itself; do not use DHCP addressing.

2. I don't understand this part of your question; more information is needed like if you are running metamod, CD, AMXX or any custom maps.