It should also be noted that what constitutes a “significant fraction” of c depends on precisely how you’re measuring the effects. You probably have equipment lying around the house capable of detecting relativistic effects from things traveling at a centimeter a minute, or so.
If I recall correctly, the only real law is that the C barrier cannot be crossed. There’s nothing preventing objects from starting out above C. How does that factor into things?
It gets around the infinite energy problem, but not the time travel problem. And that is either impossible or involves rules we don’t understand; so we can’t say what would happen, only speculate.
Tachyons are theoretical (I use that term non-scientifically) particles that do just that. They travel at FTL.
The simplest solution would be to do your FL travel in a Newtonian universe. Time would be constant in all reference frames - no need to reset your watch (except for starlight savings time).