This answer also applies to your other ongoing post about having 2 SVIs for the same vlan, one on the closet switch and one on the core switch.
Q1 - correct. On core 4 the packet could be forwared in vlan 20 at L2 and that's why i said you could have a trunk between core 4 and sw5, but it can't be routed.
Q2 - this and Q3 are rather large questions. And we need to cover a little bit of terminology. In the Cisco hierarchical design model there are 3 layers -
access-layer - pretty much where end user devices connect to.
distribution layer - where access-layer switches are aggregated to and in the traditional design where the inter-vlan routing for the access-layer vlans takes place.
core layer - basically an interconnect between distribution switches. Recommendation for a while now has been to route between the core and the distribution layer although with the
advent of VSS on the 6500 and VPCs on the Nexus switches having L2 back to the core is now a perfectly reasonable design approach.
Often in smaller deployments such as single buildings etc. you can have a collapsed distribution/core ie. they are the same pair of switches.
As i said above the traditional design was L2 from the access-layer to the distribution switches and then route between vlans on the distribution switches. So the uplinks would be L2 either a trunk or in a single vlan and the SVIs for the access-layer vlans would be on the distribution switches. A more recent development is to use L3 routed links to connect from the access-layer to the distribution switches so the SVIs would be on the access-layer switches for the access-layer switch vlans.
The reason i'm explaining all this is to try and put Q2 into some context for you and as i say this also applies to your other thread -
https://supportforums.cisco.com/thread/2003930So if the 3750 is a L2 device and has a vlan eg. vlan 10 and it connects to core 4 via a trunk connection then the SVI IP address that the devices in vlan 10 use as their default-gateway should be the SVI IP on core 4. If the 3750 has an SVI in vlan 10 it should only be used to access the 3750 remotely to manage the switch.
If the 3750 is a L3 device then it becomes more complicated.
If you connect the 3750 to core 4 with a L3 routed link ie. not a trunk, and you want to route vlan 10 off the 3750 then the SVI IP for vlan 10 on the 3750 is used as the default-gateway for end user devices. But more importantly you then wouldn't have an SVI on core 4 for vlan 10 because vlan 10 cannot exist on the 3750 and then on core4 separated by a L3 routed link.
I'm not sure exactly what design your network topology is following. There is no heirarchy among the "core" switches and although it looks like an attempt is made to isolate access-layer vlans to each core switch there are then 2 site wide vlans, vlan 13 & 253, that interconnect all your switches. As i said before one of the main problems you are facing is your network does not readily lend itself to spanning vlans across multiple core switches which in itself is no bad thing but it is when you need to span a vlan across multiple switches if you see what i mean.
Jon