Cisco 2960-X IP Address Configuration

The Cisco 2960 switch is a L3 capable switch. Today switches are usually no longer confined to L2 and the router/switch delineation is blurred from the traditional definitions.

You are missing the ip routing command from your config. If you add this command you will find that the VLAN SVIs can route directly between each other on the switch. More information on this feature can be found here: Configure InterVLAN Routing on Layer 3 Switches

However I do not see any routing configuration or default gateway configured on your switch whatsoever, so I doubt that the switch itself could reach any remote network assistance. I suspect that proxy-arp is what is being used here and the major classes are what is 'choosing' the VLAN. Proxy ARP

The non-Cisco switch will be running MST if it is using Rapid STP. You need to make sure your Cisco switches are also running MST, not the default PVST+ or Rapid PVST+. You need to run the same STP version on all the connected switches for STP to work correctly.

Once you fix all that, make sure that the native VLANs on each end of a link are the same. Cisco switches will use CDP to verify this, but your non-Cisco switch probably doesn't have CDP, so it may ignorant of the mismatch.

EDIT per the comment:

Different versions of STP use different BPDUs to establish the root switch and root ports. Cisco PVST+ (including Rapid PVST+) uses a separate STP instance for each VLAN, while MST uses one instance of STP for each user-defined group of VLANs, and, unless you take pains to do this, the two STP versions probably won't match in this regard. When mixing the two, it is normally best to use MST, and correctly match the VLANs to the STP groups. This works well, but it is certainly more work than using the Cisco STP version everywhere.

It is possible to interoperate the two versions, but it can be difficult to get it right so that the root and root ports are correctly identified on each switch in the layer-2 domain. Having inconsistent root switches and root ports can cause STP loops resulting in broadcast storms which can bring down the entire layer-2 domain.

It is much easier to use a single STP version across all your switches, and, unfortunately for added complexity, that would probably be be MST when mixing Cisco and non-Cisco switches.