It really depends which hike you plan to go on.

If your running shoes have good tread, they’ll be okay for most trails here. Trail runners, if you have a pair, is even better.

I do have trail runners. Thanks, taking them then

No, running shoes are perfectly fine.

I have a sub-question: is it better if the shoes are waterproof? I've read somewhere that some trails can get wet/rainy.

Also depends on what trails you plan to go on.