The Zeroth Rule of Time Travel: It's your world, so it's your rules. Time travel works the way you want it to work. There is no "right" solution (even given the current mathematical theories we have today).
So, what we're really doing here is pointing out inconsistencies with your rules so you can refine the rules of time travel in your world. Excellent! That's what we do here.
Cate lives to be an adult and at midnight on November 1, 2020 begins moving backward 48 hours every midnight. Our "target" Cate (call her "present Cate") has a continuous memory from early childhood through each time hop. Given enough hops (and ignoring a LOT of details), she eventually ends up flat on her back on bare ground because she reaches a time before the building exists. This is theoretically possible because she's moving backward in time at twice the rate she's moving forward in time. So, if she lives through this for 50 years, she travels backward 100 years.
Problem: How many Cates are in the bed at midnight?
Remember the Zeroth Rule of Time Travel: it's your world, you make the rules.
If you decide time is a continuous stream that coexists past, present, and future, AND you decide that Cate's time hops are the property of Present Cate and no Past Cate, then Present Cate arrives in the same space as Past Cate and a tremendous amount of dry cleaning ensues. Charles is scared for life as he must climb past offal equal to twice the body weight of Cate. Yuck. This isn't an interesting set of rules as it stops the time travel on the first hop and basically erases Cate from the time line.
If you decide time is a continuous stream AND that Cate's time hops affect all Cates past, present, and future, then there's no coexistence conflict as Present Cate arrives on the edge of time (having "spent time traveling" that'll cause a headache) just after Past Cate left. No dry cleaning, but Charles still needs counseling because the Cate he wakes up with knows things about the next 48 hours that she shouldn't. (If Cate moves far enough into the past, ignoring the fact that she's aging as she goes, she'll eventually be burned as a witch.)
This particular rule has a bit of a problem because all Cates are affected. That means baby Cate just hoped backward 48 hours — which will be a surprise to Past Dad and Past Mom, who hasn't gone into labor yet. That means this rule doesn't work well, either.
- Let's decide time is NOT a continuous stream. You can believe that each "time stream" is an alternate or parallel reality or that they simply don't exist, the only stream that does exist being the one "Present Cate" is in. (It doesn't matter which you choose because it's all window dressing, the effects are the same.) If you do this, then the moment Present Cate begins to shift backward in time all of creation that isn't the target point in time 48 hours in the past cease to exist.
This is a great rule because it eliminates all of the paradoxes and all of the problems. Cate never arrives on top of her past self and her past self, having ceased to exist, doesn't travel. This means Baby Cate remains the memory she should be and Mom and Dad don't need counseling because every 24 hours someone that looks a bit like them shows up with increasingly capable language skills.
It also eliminates the problem of what to do with each point in the past. Think of it this way. Cate starts time hoping two months in the future compared to the day Charles started sleeping in the same bed with her. From the perspective of Charles on that first night, he wakes up each morning on the same day (that first day) to an increasingly older Cate who knows more and more about what's going to happen further and further into the future — except that from his frame of reference, each time he wakes up it's as if it never happened before. Yeah, I like this rule. It's neat and tidy.
But don't simply take my word for it. You might want the ugliness of one of the other rule sets. You'll probably get a headache keeping everything straight, but that's what being an author is all about!
Will Charles ever wake up with a younger Cate?
Nope. Charles always wakes up with an older Cate. Someone from the future.
Cate, on the other hand, always wakes up next to a younger Charles — until she passes the date when Charles started sharing her bed. One of the great problems of Time Travel is what happens spatially when someone travels? This is really important because the Earth is rotating, and orbiting, and the sun is moving... From a "realistic" perspective, Cate will never appear in her bed. She'll appear in a geospatial location where the Earth was. Depending on where she started, she'll either be underground or above the ground. (The odds of her being anywhere on the ground are so close to zero that they're not worth considering.) Go back far enough and she appears in space. At that point her hops must coincide with the orbital passage of the Earth, which creates ever-narrowing windows until any more distance plops you a long way ahead of where the solar system will eventually be.
This means your time travel rules allow for both a change in time and a change in position to guarantee she's always in the location of the bed.
Except that things change. Beds move. Houses are built. You'll either need to ignore those ugly details for the sake of your story, or you have to incorporate a limit to how many hops are made, or you get to worry about every little messy problem that could occur. Heaven help Cate if the cat decided to sleep on the bed on December 31, 2019, while she was up partying with Charles for New Year's Eve. Cat... Cate... Dry cleaning...
Finally, can Cate and/or Charles change the future?
Remember the zeroth rule of time travel: it's your world and your rules. If time in your world is a continuous stream, then somehow "time" won't let them change the future — or it will and that creates paradoxes, and how those paradoxes resolve depends on which of the three earlier examples of rules I gave you (or any other rules, if you so choose) are used.
Given your initial conditions, Cate will never meet herself. But she could (for gruesome example) kill some Past Charles. Now what? Does she remember the previous-timeline-memories she had of Charles? Does Present Cate cease to exist because the time conditions that brought her into being no longer exist?
It's your world, so you set the rules. Which rules make sense depend a lot on where you want to take your story. Just remember — there is no "right" answer. It's your world. What ultimately matters is your readers: they expect your world to be imaginative and consistent.