Are there new stars appearing all the time as their light finally makes it to our solar system from the far reaches of space?

Yes, past present and future

Yes, but at a point that will not be the case as space is expanding as well. If it expands at the speed of light, then it would not be possible for light from objects in space far away to reach us.

Yes.

Not "all the time," but often.

Yes, sort of. The process of a star "turning on" is a very slow process in human terms, many hundreds or even thousands of years. It's not like they just suddenly turn on. At the same time, there are other stars going dim. So the overall about of star light from any patch of sky is pretty much the same all the time, unless there is a supernova.

It can be thought of that way. Like the nebulae around us that are producing new stars, there are those in galaxies far off likewise producing new stars. And, there are also stars that are 'dying' as well - supernovae that flash bright, then fade over time.

Any star that is so far away that its light is just now reaching us would be too small to see with the naked eye. The sky has all the stars youll ever see.

As for with a telescope, I dont know