1/ Can I hook it up to my home wifi network and backup multiple home macs's time machine backups?
Yes and no. Yes, if your "main" router is also another Apple AirPort product. No, if it's not.
2/ Can I hook onto my LAN network? DO the same as the above so its more reliable?
Yes, this is the recommended way to connect an AirPort Time Capsule to a network. The AirPort Time Capsule can be configured to provide a wireless network in this type of setup, or if you don't need the WiFi service, you can turn if off and the AirPort Time Capsule will act like a wired hard drive.
3/ Can I extend my existing Linksys Mesh with this into a dead spot wifi area ?
No, the AirPort Time Capsule cannot connect to a third party network using only a WiFi connection.
However, if you connect the AirPort Time Capsule using Ethernet, it could be configured to provide a WiFi network that uses the exact same wireless network name and password as the Linksys mesh system, in effect "extending" the network that way.
The downside........you have to run an Ethernet cable from the AirPort Time Capsule location back to a Linksys mesh router.
As far as backups, it never hurts to have two sets. If your other Time Capsule is also a model A1409, it is at least 8 years old and as much as 10 depending on when it was placed into service.