You don't need a static IP for what you've described -- just get your new router and you'll be fine.
You have two requirements:
(1) Remote access. You can either use Windows built-in Remote Desktop (free, but requires some configuration on both the client and the host desktop); or the very-easy-to-use LogMeIn Pro ($69.95/year for a single PC). I don't know why your ISP told you your IPs won't work with LogMeIn => I find that VERY unlikely. Download the free version (or the free demo of LogMeIn Pro) and try it ... I'm fairly confident it will work just fine :-) It's very simple to use ... and if you set up one PC that you can control remotely, you can easily access your network from that PC. The free version has a few restrictions on its capabilities that would make it less useful for you; and it's only licensed for personal and home use, which doesn't seem to fit your situation. Details here: https://secure.logmein.com/home.asp
(2) Potential web hosting. For this, you need a URL that always "points" to the same location (your web server). One way to resolve this is with a static IP ... but a free alternative is to use the free redirection service provided by http://www.dyndns.com/ [The link suggested above will simply redirect to the actual address, which is in the .COM domain]. This sets up a small update client on your PC, which communicates with dyndns to keep your "real" IP mapped to a host name provided by dyndns. You then access your web site through that host name. Works perfectly ... I've done this for several folks who run web servers at home to provide remote access to security cameras.
Bottom line: You almost certainly don't need to add any services to your ISP account ... what you have is plenty (you simply need a new router).
The 2nd IP they allow you is nice, but you don't really need it and aren't likely to use it (except for right now while you're "router-less". If you wanted to totally isolate your web server from your system, you could use the 2nd IP for that system => to do that you'd need a switch between the cable modem and router ... just plug the web server system and router both into the switch => the router would get one of your IP's (and share that connection with all the systems connected through the router); and the web server system would get your other. But there's really no reason to do that -- and it would have the disadvantage that the web server wouldn't be on your network, so you'd have to update it manually (you could connect via the internet).