the 2950 is a layer 2 switch only . You can create as many vlans as you want but if you want them to talk to each then you would have to trunk to a router or layer2/3 switch to route between those vlans . The ip address on this switch is to manage the switch only , the same with the default gateway . It can do no routing .
You ask a somewhat complex question with several parts that are mutually contradictory. so let me answer in this way:
First lets clarify what we mean by VLAN and what we mean by subnet and the relationship between them. A VLAN is a layer 2 concept. A VLAN defines a broadcast domain in which any broadcast frame generated by any host will be received by every other host in the VLAN. A subnet is a layer 3 concept that identifies addresses that should logically be able to communicate directly with each other. The relationship between them is that we usually assume a one to one relationship between VLAN and subnet so that a VLAN is a subnet and a subnet is a VLAN.
But that is a general assumption and is not always the case. In the first part of your question you ask if it is possible to have 3 different subnets in the same VLAN and the answer is yes you can have that. It is not generally the Best Practice but it can be done. So it is possible to have a VLAN which contains 192.168.13.0, and 192.168.14.0, and 192.168.15.0 all in the same VLAN.
However the following part of your question departs from that and contradicts that. You suggest that the VLAN interface might be 192.168.10.5 but that address does not fall into any of the subnets that you have listed. And you suggest that they would go to a gateway of 192.168.5.1 which is also not related to the subnets that you have listed.
You are correct that this is not the correct way of doing things and taking the question in all its aspects no it would not work.
I hope that my answer was sufficiently clear that to work properly the gateway address must be in the same subnet as the management address of the layer 2 switch. In the example that you suggested the management address of the switch was not in any of the subnets which you suggested would exist on the switch. And the gateway address was not in either the subnet of the management address or in any of the subnets. For that reason the gateway that you suggested would not work.
We are pleased to announce availability of Beta software for 16.6.3. 16.6.3 will be the second rebuild on the 16.6 release train targeted towards Catalyst 9500/9400/9300/3850/3650 switching platforms. We are looking for early feedback from custome...