VLANs and L3 switches

I've been racking my brain on this question for a few days now, in the standard L2 hierarchical LAN all VLANs are defined at the distribution layer and VLANs span access switches. In the Layer 3 access model, VLANs are defined in the access switches and cannot span other access switches. What confuses me is:

1. If IP routing is enabled and each VLAN has a configured SVI, if the links between 2 switches are trunks (not configured with a "no switchport" and IPs on both ends) is it doing L2 switching or L3 routing?

2. If switches are doing L3 routing (IP addresses assigned to connected switchports) how does it handle the "same" VLAN on both sides of the switch? At that point they are effectively different networks so my assumption oh how it handles them is "not very efficiently"

Bonus question. Assuming there is 1 switch acting as a "Core switch" with VLAN 1 being the native VLAN for all switches, the default route of all other switches doing either L2 or L3 should be the cores VLAN 1 IP and not the IP of another VLAN defined at the core, correct?