Yes. What you want to do is called a wireless bridge, and it is possible. Not all home routers support it, so do your research ahead of time. The free DD-WRT and Tomato firmware support this and work well, so if you get a router which supports one of those, it would work. The WRT54GL works very well for DD-WRT and can do bridging.
The way bridging works is the router acts as a wireless client, meaning it connects to the existing wireless network. It doesn't matter what kind of router set up the existing wireless network. You can then connect computers to the 2nd router's ethernet ports.
DD-WRT Wireless Bridge Mode documentation
DD-WRT supported hardware
I will assume that you are using consumer grade routers such as the ones sold by Linksys or Netgear. Please let me know if this is not the case.
Most of these routers do Port Address Translation out of the box. This means that they allow all hosts on the LAN side of the router to access networks on the WAN side of the router using only one IP address assigned to the WAN interface. This creates a problem when trying to access hosts inside the network. You would have to create static NAT rules or Static PAT rules to the inside hosts (Port Forwarding.)
In your specific scenario, you should have no problem accessing hosts on Router 1's subnet from hosts on Router 2's subnet. You should be able to ping and access these hosts using RDP. The reason this works is because the routers are smart enough to allow a connection back in as long as that connection was initiated behind the NAT/PAT.
Accessing hosts on Router 2's subnet from hosts on Router 1's subnet isn't as straight forward though. You are right, one way is to do Port forwarding on Router 2 to hosts inside its subnet (Static PAT). Some routers support more than one Static IP address on their interfaces. If your Router 2 is one of those routers you could create Static NAT rules instead of Static PAT (port forwarding). This would assign one Static IP address to every host inside the subnet that you want to access from the outside of Router 2. This will allow you to use different IPs for every host and not port numbers with the same IP. Unfortunately not every consumer grade router supports this.
Static NATs and Static PATs are the only two ways to access devices inside a device that is using NAT/PAT (other than VPNs, but why complicate things).
The other solution is to disable NAT/PAT on Router 2. Unfortunately, not every Consumer Grade Router supports this. By disabling NAT/PAT on Router 2 and just enabling Routing, you will not need to create Static NAT/PAT rules to access devices on either subnet. You can just type the IP address of the device, and you will be able to RDP or Ping to it. As far as file sharing, I always prefer to use mapped drives. You can do this when there is not NAT/MAP involved.
These two solutions are a bit more complicated, most consumer grade routers do not support those types of configurations. If your router does not support these configurations, then the only solution is to replace one of the routers.
Replace Router 2 with a Switch with enough ports to accommodate all hosts on Router 2's LAN. This will get rid of the NATting device and allow you to have all hosts on the same network. If you don't want to spend the money on a new Switch, you can disable DHCP on Router 2 and plug in the cable that goes from Router 1 to Router 2 on one of the Switch ports for Router 2 (NOT the WAN port). This will allow you to use Router 2, as mentioned by a previous answerer, as a switch.
Let me know if this helps,
Best Answer
There's some good suggestions here: How can I get the same SSID for multiple access points? Personally, I use three routers with a hard-wired backbone, all with different SSIDs and passphrases and on non-overlapping channels. Only the router connected to the internet is used for DHCP though. It's a more complicated setup but it works for me.