And when it becomes a multi-celled organism through various mutations?

If you could trace your lineage back to the first single-celled organism, would that make you the oldest lifeform on the planet?

There's a good argument to be made that organisms which reproduce with haploid gametes (animals and plants) truly generate "new" individuals when the sperm and egg fuse to create the first cell with a nucleus containing a truly unique genotype. However, organisms that reproduce by fission, where one complete living cell divides into two identical complete living cells (minor single-site mutations aside) have a much stronger case to claim that they are both still "the same" organism, as old as the original. So, no, you aren't the oldest organism on the planet, but many bacteria could be considered to be a billion or two years old.

It always amuses me how rapidly discussions about life (define life, what constitutes a different species or organism) end up at ancient philosophical discussions. In this case: https://en.wikipedia.org/wiki/Ship_of_Theseus is just a breath away! (In this case: would those cells be a billion or two years old? Have all of their atoms been replaced at this point? Does it matter?)

If the clones of individuals can accrue minor errors until they've diverged enough to become a breeding pair and still be considered the same organism. Why would the result of them melding their DNA together make them a new organism?

That is analogous to the "I've had this brush for years, and it's had a new handle 3 times and a new brush head 5 times." type idea. Safe to say that no, it is not the same organism.

At what point do you consider the bacterium dead? If it splits in two, you consider them both offspring and the parent is dead?

Well, the first of any living generation, sure.

We don't know that the first organism made it, perhaps it didn't. Or perhaps it divided many times before its line died out.