Would it be possible for a water based animal to swim with a tail that was structurally just human legs fused together?
If mermaiding is any indication, yes. Real-world mermaids are humans who wear a swimsuit that covers both legs and ends in a monofin that functions as a fluke. The monofin clips onto the feet, and the swimmer uses a motion called a dolphin kick. At various times, online stores have sold monofins and swimsuits designed to fit over them.
Could they walk if the tail was constructed into legs or would it take them too many years of therapy and not be worth it?
Milagros Cerrón and Tiffany Yorks were born with sirenomelia (fused legs). Their legs were separated surgically, and with physical therapy, they learned to walk. (No word on whether anyone in the operating room was named Ursula.) Ms. Yorks still needed crutches. Shiloh Pepin was born with the same condition, but because the blood vessels in her fused foot were more intertwined, she couldn't be given the same operation.
If my Merfolk had tails could they be separated into human legs without them having a human ancestor or sharing a ancestor with humans?
That depends on what you mean by "ancestor", as all vertebrates are believed to share an ancestor. If you just mean not sharing an ancestor within the order Primates, then yes, there are examples of other aquatic vertebrates swimming with specialized hind feet: seals among mammals, and loons and grebes among birds. If you're aiming for some measure of land agility, as your later question implies, you could consider elements of the physique of Olympic gold medalist Michael Phelps: long, thin torso, long arms, short legs, and large feet connected to hypermobile ankles. (See Phelps, No Limits, p. 67.) Then exaggerate them. But if you're looking to create mer-people, as opposed to yet an-otter animal species, you still may need a path to sapience other than that which produced Homo (tree dwelling -> prehensile hands -> tool use -> sapience).