Is it possible to know if you are travelling close to the speed of light if you have nothing to compare your speed with? This is a question that has been bugging me for some time. Let's see if I am able to make me understood!

Imagine you are travelling in a spaceship at 99.9999999999% the speed of light and you are somehow within a spaceship without windows.

My question is, is there ANY kind of experiment that would show you any time/space dilation effect as compared to being stationary? Is there ANY lab or thought experiment you could make or think that would tell you there are weird things happening with time and space outside your spaceship? I mean, would physics work differently than they would at home? Would radioactive atoms take longer to decay in a way you could measure and compare with your earth-based decay tables? Anything?

Or the only way to find out would be simply to step out of the spaceship and find out that life on Earth has gone extinct in the last couple of hours?

And another question related to the first. If you were travelling in the spaceship at that speed in the X-axis direction and you handthrew a baseball in that same direction, would that ball's mass or inertia explode asyntotically to practically infinity since you are increasing a fantastically high speed a tiny winy bit more? Or perhaps you could not threw the ball at all because of the huge mass increase being that close already to the speed of light? Or you wouldn't notice nothing special and a regular sized with normal weight baseball would impact the other side of the room at a relative speed to you of 60 km/h?
Velocity is relative. From your perspective, your velocity is zero. Without windows (or sensors) on your spaceship, you cannot know the difference. To measure your velocity, you have to define a reference point, eg a star, the average velocity of surounding stars or the galaxy center. If you have such a huge velocity relative to them, from you perspective, they are this fast and not you. So you can observe the time and space dilation of these objects, while your time appears to be normal. You ship is an inertial reference frame.

As for throwing the baseball: From your perspective, you have the speed zero, and the baseball eg. the speed 0.5c. A "fixed" observer, which observes you at 0.99c will measure the balls velocity at 0.997c. With those relative effects, speeds don't add up, only kinetic energy does. http://curious.astro.cornell.edu/about-us/139-physics/the-theory-of-relativity/special-relativity/1016-why-can-t-relative-velocities-add-up-to-more-than-the-speed-of-light-intermediate The increased mass is only a relativistic observation, resulting from the kinetic energy (relative to the observer).