This will require a "policy-based routing"-esque configuration, since you'll essentially be routing based on source address/subnet.
This should be possible with WAN balancer if memory serves, but it's been a while so I recommend looking at it's requirements first. (i.e. requires separate cards for each WAN connection)
Based on @TeunVink's comments, Problem 1 was fixed.
Below are the methods I have used for setting up the management interface.
I chose not to use VLAN 1 for management, as per many recommendations. So I cleared IPs from VLAN 1 and Management Console (Fastethernet0), as:
conf t
int vlan 1
no ip address
exit
int fa0
no ip address
exit
Assuming that VTP is enabled is working between Core and Access switches:
Create a separate management vlan in the core switch. E.g. VLAN 10
conf t
vlan 10
name management
exit
Assign IP address to this vlan
int vlan 10
ip address 10.0.10.1 255.255.255.0
exit
Then connect to the 2960 access switch via console and verify the vlan 10 is present there using show vlan command. It should be there as the VTP already would have taken care of distributing it. Now assign ip separately here for vlan 10 in the access switch as:
conf t
int vlan 10
ip address 10.0.10.2 255.255.255.0
exit
Configure ssh by following this link.
That's it. Now Ping 10.0.10.2 from other vlans to the Access switch. Should be good to go.
This link can be treated as an excellent reference for this purpose.
Best Answer
If your switch has L3 ability, you can issue the ip routing on the global config and then use the ping. Also the L3 communication between VLAN interfaces (as known as SVI on Cisco literature) starts to work. This test can be done either on the switch itself or by the hosts which are connected to their respective VLANs.
Use a extended ping command by using the source address from an SVI interface you've created towards the IP address of the another SVI on this switch.
Once these interfaces are up, and you have setup the IP addresses on the hosts belonging to them, you are able to see the ARP table for each L3 interface (SVI).
Hope this helps!