1) Will my installed apps working on Lion would keep working on OS Server?
Yes (but I'm sure someone will find an exception!)
But you don't need OS X Server to host websites - you can do it on regular OS X, which includes the same Apache web server software (you're sure to find a HowTo guide on the interwebs) and (although your mileage may vary depending on what you want to do) I didn't find the Server tools very helpful for this unless you just want a click'n'drool wiki server configured the way Apple intends. If you're out to learn, then finding out how to edit the Apache config files in regular OS X and installing any wiki/blog/CMS/whatever software you want (most are free) will be useful to you.
2) How do I know if my ip is static or not on Lion?
Check the details provided by your internet provider. Its the only way to be sure. These days static IP is usually advertised as a "feature" so if its not mentioned, and if you didn't get your own domain name you probably haven't got it. If you don't have a static IP you might want to investigate this:
http://en.wikipedia.org/wiki/Dynamic_DNS#DDNS_for_ISP_users (I've never tried it).
Also, if you're connecting via a router/firewall (likely if you've got two machines) then its your router's IP address that matters - in that case your macs will have private IP addresses (10.x.x.x, 192.168.x.x, 172.16.0.0 - 172.31.255.255) only valid on your home network and you'll have to set up
port forwarding on your router.
3) Does my system should be running all the time to keep the websites up?
Yes.
The old PPC Mac Mini that handles my mail has been running continuously for several years - currently 312 days since last restart
(Sadly, its running Linux, not Mac OS).
4) And Do I have to talk with my internet service provider about doing this. (they won't block my ip?)
You should check your "acceptable use policy". Some providers forbid running servers. This isn't just being mean: ADSL connections are designed on the assumption that you are going to be downloading far more than you are uploading, so the upload speed is only a fraction of the download speed. If someone else downloads from your website then they're using <i>your</i> upload bandwidth.
So I'd echo what others have said - while its OK for creating experimental sites that only you will use, using a home broadband connection to run any sort of public website that might attract traffic is a non-starter: that's what web hosting services are for.