Would it be feasible for them to continue living,
This is actually the most feasible outcome! It requires the least amount of magical effects.
There is a mode of time travel that isn't discussed above: The future is obliterated. Imagine the time machine resets the entire universe to a previous state, with the only difference being the machine itself, and Bill inside it, are protected from this reset. (for discussion, call this TT-Bill, for Time Traveled Bill).
That becomes the equivalent of a machine and person(s( from the future simply materializing, in an instant, in our time. TT-Bill's time and what led to TT-Bill no longer exists. The universe is reset from 2342 to 2017 with a simple edit: a cut-and-paste of a sphere with a ten foot radius. Everything inside that sphere is severed from cause and effect, TT-Bill's 2342 universe and all the events leading up to him, his mind and knowledge and equipment, cease to exist, but the product of all those events, the physical manifestation of TT-Bill and his machine, is preserved (and transported to wherever in the Universe Earth was in the past; since Earth is looping its way through space).
You can let TT-Bill travel into the future in a matter of seconds, as far as you want; this is just Einsteinian relativistic time dilation, with a little hand-waving to stay in place. Any effects TT-Bill had on the past just evolve naturally for however many years or centuries are desired. So anything TT-Bill did in 2017, including exploding nuclear bombs and killing dozens of his own ancestors (or all of his own ancestors) has its effect.
In the new version TT-Bill is never born, but it doesn't matter: That was true the moment TT-Bill appeared in 2017: the future was dissolved. Now, just like us in real life, every tiny action or decision TT-Bill takes has an effect on the new future. But what happens in that new future does not affect TT-Bill's body or mind back in 2017; there is no thread of causality running from the future to the past, just as what is going to happen to us, in real life, does not change our bodies or mind right now.
This version of time travel also means duplication of people. Let's have a different Time Traveler, TT-Chuck, he travels ten years from 2017 into the past (2007), the younger Chuck is there. TT-Chuck is there, as a separate and real person. They both evolve; could embrace each other, speak to each other, whatever any two real people could do. TT-Chuck could kill his earlier self, or vice versa, or TT-Chuck can leave his earlier self completely alone.
However they both evolve, ten years later they are both there. If TT-Chuck travels into the future, they are both there; although Chuck has had 10 different years of experience.
For example, say TT-Chuck does something to prevent the imminent accidental death of his (in 2007) 12 year old girl, successfully. That 2007 world is now different than the one TT-Chuck experienced. Let TT-Chuck jump 10 years into the future: Chuck is now the same age as TT-Chuck, but never experienced the death of his little girl: What happened is Chuck came out to drive her to school one day, and found a flat tire with a nail in it. TT-Chuck hammered it in at 3:30 AM, and made sure the tire was completely flat, but Chuck was just irritated: He spent 45 minutes changing the tire, drove his girl to school and made excuses, and that is it. Ten years later, he barely recalls the incident, and neither does his girl.
Chuck was never in the car accident that killed his girl. The rough-housing speeding teens that crashed into them were not killed either. TT-Chuck still remembers it all vividly, the accident, waking with his girl dead and staring beside him, his inability to revive her. The funeral. His emotional scars all remain intact. Nothing is faded.
Because TT-Chuck's brain and neural patterns and connections were transported from 2017 to 2007 intact, a physical system of synapses and neurons, that are not magically affected by what then transpires. New memories (or injuries) can be formed (or suffered) but that isn't magic, that is normal biological processes. Whereas, it would take some form of magic to undo such physical systems and replace them with some other system.
So if he likes, he can try to find a way to watch his little girl grow up. With some preparation from 2017, he can become nearly instantly wealthy in 2007. The changes he intends to make won't have much or any effect on the stock market or Vegas betting markets for sports games. In 2017 he can find convicted forgers that were active in 2007; and get a forged identity he can use in 2007. He can buy the house next door to Chuck and remark on the incredible resemblance between them, they must share some ancestry! TT-Chuck can become lifelong friends with Chuck (he obviously knows enough about Chuck to do that); so he can watch him raise their little girl. And if necessary, time travel again to save her again; creating yet-another Chuck. Or if the proliferation of time traveling Chucks becomes an issue, send a video from the future explaining everything to the first TT-Chuck, including how to prevent the problem: The TT-Chuck sending such a video dissolves; the TT-Chuck receiving the video checks his time-machine's messaging system every day for precisely this reason; and knows it is the real deal.
All other systems of time travel, characters fading out (like in Back to the Future), vanishing, or the universe preventing paradoxes, require some kind of unexplainable magic effect. So does this "reset" version of time-travel, but it is an instantaneous effect that is not operating in any continuous manner or causing any observable and unexplainable physics. One second there is air, in the next second the air is pushed aside and replaced by TT-Chuck, with his very high-tech Time Traveling device surgically implanted as a marble-sized object in his sinus cavity. Then the world, physics, biology, life and death all proceed normally.
TT-Chuck cannot help but have some effects, he is an obstacle to breeze and sunlight, he is consuming oxygen and weighs a few hundred pounds as he walks on the ground. But there is effectively zero chance such effects "butterfly" into any significant changes to the future in the next few years. Heck, 99.9% of normal human deaths have no measurable impact on the course of stock markets, business, politics, wars, sports, or anything else.