How to connect via static IP to a web application?

Question

Computer networks

Hello! The situation is this:
Yesterday I deployed tomcat 7 on my machine, put my application on it.
Through the localhost address, I connect to the application perfectly.
I bought a static IP from an online provider. I noticed a strange thing - in the personal account of the IP provider one is written, and services like myIp show another.
I used the one that the myIp service gave me. The first time I could not connect, the baruser hung-hung and no effect. After some time I managed to get in.
This morning I start the computer again, but I can’t connect - the baruser hangs and nothing responds. It's been like this for a couple of hours ...
Hence the questions:

• Why does the provider think that I have one IP, but myIp services show another?
  
• Why did the server work by itself? Does he have to register somewhere on the Internet, or what?
  
• How do I properly deploy a server on my machine, maybe I'm missing some nuance? The port on the router seemed to be forwarded, and yesterday everything worked!
  
• What are the means of monitoring (debugging?) Connections?
  

I ask you not to scold too much, I'm a programmer, not an admin.

Answer 1

Colleagues, the casket has opened.
The OnLime provider tied ip to the MAC address of the wrong device (I also have an IP telephony gateway from them), and therefore the IP was not the same.
Now everything has been corrected. Thanks for the advice, colleagues!

Answer 2

How is your internet connected? Directly to the computer?

Answer 3

Or maybe it's that the router through NAT just changes the IP address?