Netgear WNR1000 Router

You're right. It's because you're on a separate network (the one your router creates) and a separate subnet. You can configure your router to act solely as an access point, by following the instructions here. Of course this will disable DHCP on your router, and if your router is on the RG in DMZ+ mode, you'll have to choose one or the other. 

Not normal. Here's a few things to try,  A) Reboot both the RG and router. B) If you have a spare ethernet cable, try connecting it to one of the ports on the router. (Don't use the one from the RG to the Router, because your router will not be able to assign you an IP Address -- DHCP is disabled.)  C) Did you do a reset of your netgear router prior to performing the process? If not, reset and repeat the process. (It would rule out any rouge misconfiguration.) 

If you plan on having your Residential Gateway and Router and roam on them, you'll have to set them on different channels. For example, it is likely the RG may be on channel 6. You'd want the Netgear on 1, 3, 9, or 11. (Some people claim poor performance on 1 and 11.) Note the channels you have them on, as this is the only way to identify which access point you're on.

For Windows 7 or 8.

If you have something like inSSIDer Home, you're pretty much set on figuring out which channel you're on, as well as finding out what channels are least cluttered. If you don't have anything like inSSIDer Home, you can use Network Shell to figure it out. 

To use Network Shell

1. Press the Windows Key + R
2. Type CMD and press enter.
3. Once CMD opens, type "netsh" and press enter.
4. Type "wlan" and press enter.
5. Type "show all" and press enter.
6. Scroll up to find the following section

   =======================================================
   ==================== SHOW INTERFACES ==================
   =======================================================
   
   
   There is 1 interface on the system:
   
    Name : Wi-Fi
    Description : Your Wireless Card
    GUID : 
    Physical address : 00:00:00:00:00:00
    State : connected
    SSID : YourWiFiSSID
    BSSID : 00:00:00:00:00:00
    Network type : Infrastructure
    Radio type : 802.11n
    Authentication : WPA2-Personal
    Cipher : CCMP
    Connection mode : Auto Connect
    Channel : 6
    Receive rate (Mbps) : 300
    Transmit rate (Mbps) : 300
    Signal : 94%
    Profile : YourProfileName
    Hosted network status : Not available

    Your channel will be shown here. 

    

   If you're running Mac OSX

   Press and hold the option key and click on the WiFi icon. The channel will be displayed along with other information regarding your connection.