When a packet is sent to the IP stack, the driver searches your routing table for the route that best matches the destination IP address. It does this by performing a binary and of the destination defined in each route and the destination defined in the packet. Whichever result most closely matches the destination defined in the packet is the route the driver uses. If there is a default gateway defined for that route then the packet is sent to the gateway. If there is no gateway defined for that route then it is assumed that the destination is on the local network and the packet is sent directly to the destination.
Thus, a packet with a destination address of 10.1.1.67 will match the route 10.1.1.0/24 and will leave on port eth0 as configured by the second route. A packet with a destination address of 10.0.0.84 will leave on eth1 as configured by the first route. Any other destination address will only match the third route and will leave via eth1, headed for the gateway.
There is no way to say which route is "closer" to the internet as none of those routes deal with publicly routed addresses, rather those are all Class A private addresses. Class A, B, and C addresses ranges that are reserved for private use and are not typically routed to the internet. It is entirely possible that the packets will be routed through other networks before reaching the internet or they may never reach the internet at all.