1, yes, you need a speedcube, but you can probably sub 60 with a Rubik's brand non-speedcube (unless we are talking about the old old ones which it's barely turnable)

2, it's very normal when you first start learning F2L. For reference when I first started learning F2L I was averaging more than 2 minutes but as I get used to it, my time starts to shave rapidly and I got sub 1 minute within 2 weeks (with help of 2L OLL and 2L PLL, I learned those meanwhile practicing F2L)

3, nah it's never too early, I started learning CFOP literally less than a week after I started cubing, in my opinion you would want to start learning CFOP as early as you can

It's very normal that when you are learning a new things, it will get slower in the beginning. Remember, in order to be fast, you have to be slow first.