In today's world, router is kind of an "all in one" device. My room Router B currently hooked up an ethernet port from Router A.
I am correct to say that there are two possible outcomes?
If Router B is behaving as a router and creating a new subnet, will it be able to access the home media server?
Router B is forming a new subnet
Technically, routers don't form subnets. All you need to make a subnet is a couple machines that are set to same subnet mask and connected to the same switch - you don't really need a router if you don't want this set of machines to talk to external networks or the Internet.
It's completely possible (but not usual at all without VLANs) for more than one subnet to be on a switch. DHCP is often used to automatically set computers on a switch to the same subnet - DHCP settings plus the IP/subnet setting of your router's IP is what "forms" the subnet in most instances.
Router B is acting as a switch (no routing functionality at all)
DHCP tells systems what the "default gateway" is - and this is what IP address that systems will send traffic to that is destined for the Internet. If no DHCP server is giving the second router IP out, and no system has its default gateway set to the second router IP, nothing is going to use it.
Connecting 2 switches (whether standalone or part of a router's "LAN" ports) enables all systems to see each other if their subnets match. The router doesn't need to be told anything for this to work, but the router's DHCP server should if you don't want to manually assign IPs and subnet masks.
No, you don't want two DHCP systems on the same switch or set of connected switches (unless they are working together in a failover configuration)
If Router A is connected to home media server, should I set up my router as a switch or as a router?
It's simpler and easier to simply use the switch of router B and disable DHCP/routing (you may not be able to disable routing but should almost always be able to disable DHCP) on router B.
Yes, you can connect a LAN port of router A to the Internet port of router B and make it so that any access to the systems "behind" router B need go through router B. Most routers support DHCP through WAN or Internet port as that's how a lot of cable companies work (Comcast assigns your router's WAN interface an IP using DHCP, for example).
You may want this if
you want to protect those systems from router A's LAN using router B's firewall functions.
you want to make it quite difficult (not impossible) for a system from the Internet to originate a connection to something behind router B.
This is typically overkill for typical home use.
Dynamic routing
This is used only if the number of routers on your network can change over time. It lets routers find other routers. You really only want 1 router (doing routing/DHCP) on your network if possible.
For the most part, you are correct on your outcomes. It's important to note that Router B will always only consume the maximum bandwidth of the port to which it is connected and its uplink port; if you hook up a 10Mb/s router to a 1Gb/s port, everything connected to the slower router will be limited to 10Mb/s when reaching the internet. Every network chain is as strong as its weakest link.
The default for Router B will be to create its own subnet. If it's wireless, this will extend to its built-in access point. Devices not connected to this router may have issues reaching your media server unless you specifically set up the subnet router firewall to permit the connection and you may have issues with network discovery on some devices. You will also want your two wireless networks to be on different channels to prevent interference with each other. While it may be more difficult to set up or adjust later, I recommend making the second router into a dumb switch and turning its AP off or allowing it to be an access point (depending on range from Router A). This will save you some network discovery headaches, particularly on devices which use a proprietary OS like Steam OS or the kind of thing you see in a Blu-Ray player.
