Is there a way to get about a 10ns Timer resolution from an AVR CPU?
There is nothing you can do on the AVR processors that I know of, which can run on from a quite slow clock, and I think up to 24 MHz. Since your Uno / Leonardo / Micro run at 16 MHz that would be only an incremental speed improvement (to clock them at 24 MHz).
Your idea of bit-banging is interesting, but in general bit-banging is a last resort when you need extra things the hardware doesn't provide (like an extra SPI port) and in general will be slower, not faster, because it involves code rather than inbuilt hardware.
Running at 16 MHz each clock pulse is 62.5 ns which means that is the absolute most resolution it could count. I don't see why you can't count at the rate of 62.5 ns (rather than 125 ns), however that isn't 10 ns resolution.

Or, is there a stand-alone analog or digital IC that is able to read at 10ns intervals pulses that an Arduino can process?

Very likely. There are probably dedicated counting chips that would do that for you. I haven't looked, but it is the sort of thing there would be a market for. Such a chip would probably use I2C or SPI to communicate, which is easy enough to interface to.