The best practice to diagnosing a problem is trying to find the source of the issue. In this case the boot issue with the Pi 3 was actually a network problem which explains why you could not successfully ping it.
Here is my suggestion on how to get the Pi 2 to boot. Use the power supply you used with the Pi 3, since you know it works. This should eliminate one problem from the equation. Secondly connect it to the screen using the same equipment as you used with the Pi 3. Just to confirm it really is a boot issue.
The solid green LED indicates that no boot code was executed. This means you need to re-flash the SD card with the OS. I'm going to walk you through the steps on how to do that for installing Raspbian and if you have a Windows computer:
- Insert the SD card into your computer and open File Explorer (for Windows).
You will notice that the full size of the SD card is not shown. This is because Windows can't read Linux Filesystems.
- To return the SD card back to its factory settings, you first need to return the partitions on the SD card back to their original configuration. To do this use a program like MiniTool Partition Wizard to delete all of the partitions on the SD card and create one Primary Partition with the original Filesystem that is the size of the SD card.
- Open up File Explorer and right-click on the SD card and click Format. Then choose the Return to device defaults option.
- Your SD card should now be back to it's factory settings.
- To install Raspbian download the latest .img file. The current version is downloadable as a .zip file. Therefore the file needs to be unzipped before it can be used. Then use something like Win32DiskImager. Select your SD card and the Raspbian image. Finally, choose the write option. Do not disconnect the SD card until the software says the write was successful.
Your SD card should now be good to go.