Actiontec v1000h is also called gateway. In a more general term, it is modem as on the copper xDSL high speed lines.
And the ONT is more like a signal converter that converts the optic fibre signal to the electric signal that goes to Actiontec.
In terms of your first question, it is not doable: connecting your router to ONT directly will not give you the public IP address you want.
However, if you connect your own router to the LAN port on Actiontec, and then configure Actiontec properly, your router should be able to use the modem (Actiontec) WAN IP address as its own. The modem (Actiontec) WAN IP address is a public IP that can be found from the Actiontec GUI, for example, 142....
Please follow the steps below to configure the Actiontec gateway from any of your computers that is connected directly to the Actiontec network either through wi-fi or ethernet:
1. Make sure your router is connected to an Actionac gateway LAN port;
2. From your computer, go to 172.16.1.254 which will give you Actiontec GUI;
3. From the top right hand, there is a Password field that is supposed to be the default 10 character system password located at the bottom of the Actiontec gateway box. If you have never changed this login password, that default system password should let you login to Actiontec interface.
4. After you login, you will see a list of all the devices that are currently connected the Actiontec. One of those devices should be your router and router should be with a private IP like 172...
5. Next, click on Firewall.
6. From the left side, click on DMZ Hosting.
7. Go through the 3 steps under DMZ Hosting: Enable - Select a device (your router) - Apply
Your router should now be accessible by the Actiontec WAN IP Address (public IP) located at the top left side of the home page of the Actiontec GUI.
(This bypasses the firewall on Actiontec and exposes your router directly to the external network. Actiontec WAN IP address may change once Actiontec gets rebooted -- static IP won't change. So you need to know what you are doing.)
Hope the above will get you started toward the right direction.