Is it true that there are no new cells created?

Yes, and no one really knows.

It's the same as conservation of matter, all cells are from preexisting cells. You can trace a child back to the cell former from the dad's sperm and the mom's egg. Each generation can be traced by the same way all the way down to the monkeys we have supposedly come from.

Although all cells come from existing cells, the new cells are those made from mitosis and meiosis. If you have more cells then you, really have new cells.

Think of a egg that is fertilized, although both cells (egg and sperm) come from existing cells, they are new cells with different DNA than the old cells. The egg unfertilized is haploid, but becomes diploid after the sperm fertilization. So in essence it is a new cell.

No, cells do not usually spring into being. But it must have happened more than once in the earliest stages of evolution.

technically true. We (humans, plants, animals, etc) are all the descendants of the first cell that evolved billions of years ago.