Do two computers always require an IP to communicate when they are communicating via ethernet card or wireless LAN card? Are there any cases when two hosts don't require an IP address to communicate? (assuming they are using network interface cards for communication)

They need IP, because the upper layer protocols like FTP or HTTP (which you need for your applications) do not talk to Layer2 (ethernet in this case), but can only talk to IP. If you can write your own application that can talk over ethernet, then there is no need for IP.

On Windows machines, computers that are set up as DHCP clients but do not have access to a DHCP server will automatically assign themselves a https://en.wikipedia.org/wiki/Link-local_addressIP address in the 169.254.0.0/16 range.  This is the default setting in Windows and can be disabled.   That means you can indeed use a crossover cable to hook up two computers to transfer files without statically assigning IP addresses. It is probably quicker to just set the IP address on each machine rather than wait for DHCP to time out, however.

As others have pointed out, the answer is 1)  Theoretically, no IP address is needed 2)  Practically, an IP address is required In theory, you could use an alternate protocol, or even just straight Ethernet frames.  However, there are no programs that will do this and the APIs to do so on any computers you're likely to use are restricted and require a fairly deep knowledge of networking.  If you're doing network protocol research, then it's possible you may be using a non-IP protocol (possibly your own) to transfer data between two computers.  Everyone else leverages the huge amount of legacy IP software available. P.S.  The above applies to computers in  2014.  The answer would have been different in 1984 when TCP/IP was still competing with other protocols (DECNET, Novell IPX, SNA)  and may be different in 2044.

Historically, there are many types of data that have been sent in Ethernet frames. This document from 1998 called http://www.cavebear.com/archive/cavebear/Ethernet/type.html has a good set of information current to that time. RFC 7042 is http://tools.ietf.org/html/rfc7042 which is a more current set of assignments for some of the ethernet assignments under IETF control. You can also look at http://standards.ieee.org/develop/regauth/ethertype/index.html from IEEE which will lead you to http://standards.ieee.org/develop/regauth/ethertype/eth.txt which is a current edition (updated daily) of the aforementioned 1998 document. This does not address your question specifically re Windows, but just gives you some idea of the range of things that are not IP that have been done via Ethernet.

In your scenario, your PCs are attempting to attain an address via DHCP, and after that times out they assign themselves an Automatic Private IP Address in the subnet 169.254.0.0/16 so that they can talk to themselves (this address range is defined as a link-local subnet). So, they're certainly using IP to talk to each other, however they also need to use mac address

If you take two Windows PCs set fo DHCP (the default), they will both get an address assigned by APIPA http://www.pcmag.com/encyclopedia/term/37858/apipa. They will then both have an IP address, and will use NetBIOS to talk to each other, over IP. Since you didn't set an IP address, you may think they are using non-IP to talk, but the way it works, it will use IP by default, even if you don't set an IP. Though, if you manually set IPs on different networks for the two computers, then connect them, they won't talk.  That should "prove" to you that it's the IP that makes it work.