Is it OK to wash other things than dishes with the same sponge scrub?
Yes, it's ok to wash other things with the sponge. You can easily use it for counter-tops, the sink, stove top, the floor - anything like that wthout a problem.

The sponge will become dirty, of course, as you clean with it, but the same is true of washing dirty dishes with it.  If you can clean a sponge well enough to keep using it on dishes until it's soft and thin, you can clean it well enough to also use it on counters, sinks, and so on, the cleaning process is just the same.  You will find the sponge is worn after fewer dishes, specifically - but the absolute amount of cleaning should be the same, just the extra wear and tear happened while cleaning other things.

While you can use the sponge to clean most things without issue, if you're using the sponge to wash dishes and food preparation surfaces you should be very careful if using it to clean messes that aren't people-safe, in case the sponge should become contaminated with enough non-food-safe substances to leave a residue.  The aforementioned sink, counters, stove, and kitchen floor and in fact most household areas should all be fine, they are surfaces that you come in contact with regularly and should not leave anything on the sponge that regular cleaning won't take care of.  But you might want a separate sponge if you're regularly using it for things like paints and solvents, mechanical lubricants, motor oil or adhesives, heavy chemicals, pesticides, or anything of that nature - that is, things you wouldn't allow in your living space without extra precautions.