I'm new to the All-H3-CC but after reading their forums and guide I was able to get all my boards up and running. I may be able to help.
Does your board LED powers on when you plug it in?
Did you try another AC adapter with the same specs? Output 5V 2.5A
Which image did you try to load?
Did you use Etcher or Rufus to flash your image?
I know it can be a bit frustrating. Try the below steps and lets see if we can get your board working. Everything below is an Windows environment.
Make sure your memory card is an SDHC class 10 at least 8GB.
Download Rufus from https://rufus.akeo.ie/. This program is an executable .exe so no installation is needed. Just make sure you run it as Administrator.
For your 2GB board download this image. http://share.loverpi.com/board/libre-computer-project/libre-computer-board-all-h3-cc/image/ubuntu/Armbian_5.45_Tritium-h5_Ubuntu_xenial_next_4.14.44_desktop.zip If you use the wrong image it won't work as the 2GB board is using the H5 64bit. This may take a while, the image is about 512MB while compressed.
Make sure you have Winrar or Winzip install and uncompressed the image. Your image will be about 2.6GB so be patient.
When done right click (Rufus) and select (Run and Administrator). Make sure that your SDHC card is in your PC. Rufus will select the drive for you but make sure it is the correct one before continuing. Click, (Select) and find the image you uncompressed, unchecked (Quick Format) and then select (Start). This will take about 3 to 4 minutes if you are using an 8GB SDHC Class 10 card, longer if is larger.
Once the image is done click (Close) and eject your card from the PC. Inserted the card in your board. Plug your HDMI cable, keyboard and mouse and power your TV on. Make sure your Input in your TV is where you connected your HDMI cable and finally plug your power cable. Allow about 30 seconds to a minute for the image to show. If it doesn't work repeat the steps again.
The image is a desktop Armbian UOS so follow the steps on the screen which are require to get to the desktop.
Best of luck and let us know if it works.
I'm sorry to hear that you are still having issues. As I did mentioned before I'm new to this but after much trial and error I was able to get all of my boards working. I really don't think that you may have a defective board but without me having it I can't really say.
I will be posting sometime by the end of this week if not sooner some videos with all of the steps in youtube. Please note that I have no association with loverpi forums, all H3 boards, etc. as I see that many are having issues loading their images. I'm a hobbyist looking to learn like everyone else.
Once I complete the videos I will post a link here. Hope this helps you and everyone else.