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If “I” traveled into the past, would I take with me the memories of the future that I had already spent time in?

No doubt that you would, but, would you survive the trip to the past? Perhaps not, but, mass and energy would. It could be that, bodies larger than the smallest of particles can't travel to the past, so, you wouldn't survive and neither would your knowledge. We may think of ourselves as large entities, but, in the larger sense, we're just particles and energy, and it's those of mass and energy of the past that are "able" to interact with the present.

If it were possible to maintain the shape and characteristics of the human form, you'd be changing the past that you arrive at, which also means changing the future, with your "future knowledge", if you decided to use it.

As far as can be told, nobody has ever traveled to the past, and it may never happen. If it were possible, we'd know about it by now. That's not to say that the past doesn't interacts with the present universe, since, it's known that particles do exist in many places and times "at the same time".

Does matter need time in order to exist? Many years ago, my young son asked me, “what is a rock”. Being a young father, I could not just say, “I don’t know”. So I told him that a rock is a frozen moment of time.