Howdy Ace,
Welcome to the forums. In terms of gameserver hosting, most of the time I've found the hosting provider's network to be the most important part of the equation. Make sure you find a provider that has the best latency and network to your users. The difference is night and day between a server with a budget and unstable network, and a server with a reliable network (also your users/clients will be really annoyed and angry with you if the server lags or just drops traffic during peak times). Something that you don't want is random packet loss during peak hours and optimized routing to you and your users/clients. Especially with a FPS game like Call of Duty 4, latency plays a major role in the gameplay and experience. I'd suggest you contact providers near your location and ask them for a test IP and/or a looking glass and see what routes it takes to you and back.
Teamspeak servers are fairly lightweight. The only worry is that sometimes (for like OpenVZ VPSes) teamspeak could potentially trigger a "flooding warning" due to the amount of packets the default codec and configuration generates. You might have to tweak the configurations a little if this is an issue with your provider.
I have no experience with call of Duty 4 game servers, however, I'm fairly certain most game servers are single core or dual core at most. While this link
lacks a ton of details on the requirements, it very much depends on how you're planning on configuring the game servers. Like are we talking 50 players on a single game server? Or something like 10 people total? Regardless, most people usually suggest higher clocked CPUs (such as around 3.0 GHz+ CPUs) for game servers due to the large volume of CPU computations required for a popular game server. An E3 or an i7 server could be considered more beneficial here.
In terms of RAM, again depending on how many slots you're planning on having, I'd probably allocate roughly 1 GB to 1.5 GB (to be on the safe side) per game server (per every 15 person slot). However, the link I mentioned above suggested around 500 ~ 600 mb per game server.
Personally, I'd recommend a dedicated server with SSDs due to the amount of resources a high-volume game server will use up.
This link suggests that a bare minimum of 20 Mbit+ guaranteed for a 32-slot COD4 Server. Teamspeak will tell you the maximum amount of bandwidth used per person depending on your codec and quality configuration so you can get that information from there. Do some math and you'll figure out the maximum amount of bandwidth you might use when all services are full (Full Teamspeak server + Full COD game servers). You should probably start searching what you need depending on that.
While most people do want to save the most amount of money, I would still recommend spending a little extra and going with a provider that you know is absolutely reliable. Going too cheap/budget wise and later finding out their network isn't reliable could make things difficult for your community/group as you'd have to switch IPs to a different provider's. Minimize this issue, pay a little bit more and go with someone good.
