Is it okay to use USB 3.0 cable with USB 2.0 wall adapter? I bought an LG Optimus One phone a long time ago. The phone came with a USB 2.0 cable + wall adapter. 

3 months ago, I bought a Galaxy Note 3. The phone came with a USB 3.0 cable + wall adapter. The wall adapter stopped working, however the 3.0 cable still works. (When I connect my Note 3 to the computer using the 3.0 cable, it works. But when I try charging my Note 3 using a wall outlet, it doesn't work).

My question is:

Since the wall adapter which came with my Note 3 stopped working but the 3.0 cable still works, is it okay if I charge my Note 3 with the USB 3.0 cable while the 3.0 cable is plugged into the wall adapter which my LG Optimus One came with?

Note: I didn't try it yet because I was afraid it might ruin the 3.0 cable.
USB devices are designed to use a wide variety of voltages ranging from 500mA to 1500mA and have been standardized for quite some time. You can generally plug any USB device into any USB cable and into any USB port, and nothing will blow up. Using a more powerful charger should speed up battery charging, while using a less powerful one will just charge slower. A good resource for this topic can be found here.

As noted in that article, the only real concern is when charging an older phone with a newer charger, as the battery might not be able to handle the higher current. In your cause your using an older charger with a new phone, so your all good.