Are there any way to use block/item registry without creating 2 tabs? I'm one of those who don't enjoy creative tabs overflow and wish to be able to use my own tab or at least one instead of two.

Maybe it's better to ask if I'm able to use it without registetring my mod in Core.

https://github.com/Modbder/DummyCore/blob/1.8/DummyCore/Core/Core.java#L129
See that boolean in the function args? Just call that function with that boolean set to false and DC will not add 2 tabs.
You could use DC without registering your mod as a DC mod, you will just be missing some functionalities(like DC's particle system, UpdateChecker, etc)

I saw that flag and when set it to false Minecraft crashed with NPE caused by this:
https://github.com/Modbder/DummyCore/blob/1.8/DummyCore/Blocks/BlocksRegistry.java#L49
It tried to get lable from null tab. I think you should insert some null checks 😉