Your version of Eclipse is 64-bit, based on the paths and filenames.
However, the version of Java that it's picking up is 32-bit, as indicated by where it is coming from, on this line:
-vm C:\Program Files (x86)\Java\jre7\bin\javaw.exe
Program Files (x86) is the folder where 64-bit Windows places 32-bit programs.
Program Files is the folder where 64-bit Windows places 64-bit programs.
This can happen when a system has more than one JVM installed, as is often the case on Windows 64-bit (for example, the JRE download page uses the bit-ness of the browser to determine what bit-ness download to offer you, and many people use(d) 32-bit browsers even though they run 64-bit Windows).
The best way to fix this, assuming you do in fact have 64-bit JRE or JDK on your system, is to specify in eclipse.ini exactly which JVM you want it to use. The instructions are detailed in the Eclipse wiki page, but basically you have to specify the -vm option in the ini file - make sure to read the wiki page carefully as the format is very specific.
Specifying the JVM path in eclipse.ini is strongly recommended because doing so isolates Eclipse from any potential changes to your system PATH that some program installers might make (I'm talking to you, Oracle!).
Another option would be to download and use 32-bit Eclipse instead of 64-bit, but it's still strongly recommended to specify the path to the JVM in eclipse.ini.
Left for historical reference:
To check your version of Java, run
java -version
in a console (command prompt). On Windows 7 with 64-bit Java 6 I get:
java version "1.6.0_27"
Java(TM) SE Runtime Environment (build 1.6.0_27-b07)
Java HotSpot(TM) 64-Bit Server VM (build 20.2-b06, mixed mode)
Note the 3rd line, which shows that this is a 64-bit version.
On a 32-bit version you'll get something like:
Java HotSpot(TM) Client VM (build 20.1-b02, mixed mode, sharing)
If you are on a 64-bit machine, then you can install the 64-bit JDK and uninstall the 32-bit one. For instance on Windows 10, just go to Settings and under Apps, you will find Java. Click on it and you will find all the different versions. Now you can select which one to uninstall.
When the Nexus 7 is plugged in there is a persistent notification that indicates "CONNECT AS / Media Device (MTP)". In this state adb devices will not show the Nexus, or undoubtedly any other device. Not exactly obvious, but if you select the second option "Camera (PTP)" the device is available for debugging (the lesson is ignore the camera, and focus on the protocol PTP).
This configuration is persistent, and I'm guessing that with a brand new device it will connect as MTP until told otherwise.
Thanks to @Ciaran Gallagher
Settings --> Storage --> Top Left Option (Computer USB Connection) tap--> choose MTP
Best Answer
This page seems to have instructions for setting up that phone with the SDK, and for taking screenshots as well. Note: I have not tested these instructions as I do not own that phone.