Connecting optical modem to my cat5 wired House

You can connect up to 4 of those remote Cat5e jacks to the Gateway via the 4 yellow LAN ports on the back of the BGW 210 Gateway.

If you need more than 4 to make all your outlets live, you will need to purchase a switch with sufficient ports and connect one of its ports to a Gateway LAN port, and then the other ports on the Switch to the various Cat5e.  Depending on where your Gateway is located vs. where all those Cat5e cables are terminated, it may make sense to use a switch even if you only needed 4 ports.

Ideally you'd install a patch panel to terminate those cables into a nice row of RJ45 jacks (something like this https://www.monoprice.com/product?p_id=7253).  If you only have one cable to where the Gateway is, then you do have an issue in that you need both the connection to the ONT to the Gateway and get the Internet traffic to the rest of the house from the Gateway to the patch panel.  Multiple options exist:

1) Move the Gateway (as my thoughts recommends)

2) Pull another cable from where the Gateway is located to the panel

3) Use a pair of Switches that implement VLAN with tagging to manage traffic over the single cable.

The drawback of #3 is, unless you use switches that run the cable at least twice as fast as your Internet connection, you will lose bandwidth.  Switches faster than 1 Gbps are fairly expensive.

Regardless, to make all your outlets hot, you'll need a switch in that panel.

My Gateway is in my study and the Cat5e cables terminate in my bedroom closet behind a wall plate.

Yes, the ONT is on the or near closet panel?

You would need to have the gateway moved to the closet, RJ45 ends placed on the cat5e wiring.

Depending on layout expect to purchase and install mesh system for home Wi-Fi as the gateway Wi-Fi will be at the closet. 

Yes. The ONT is inside my garage and the cat5e cables terminate in the bedroom closet at the opposite end of the house. My study and (gateway) is in the middle between the two.

Sounds like option 2 would be best in my situation.

Using option 1, the bedroom closet would be a problem with too much traffic congestion in my wife's closet. It could almost work if the  BGW210-700 wasn't so bulky. If I could use something smaller in the wall panel (there is an electrical outlet) along with a patch panel. But that may be a problem being enclosed causing it to get too hot.

I like these for 12 cables or less says CAT6 that's just the rating you can punch down CAT5 on them.