If you can prove the offer of $19/hr after probation then you have legal standing to sue them for the different between what they are paying you and the $19/hr. If and when they specifically tell you that you will be making only $15/hr from that time forward you will no longer be able to sue for more money but you can still sue for the hours you should have made $19/hr. That being said you will then have to determine if that difference is enough money to make legal efforts and the damage to your relation with your employer worth while

You can quit and find a new job.

You can IF there is something in writing.

No. You have no right to a raise.

You can legally quit and find a new job. Unless you have a written contract stating you would get $19 an hour after the probation period then you have no course of action.

Take what you have in writing and ask when you can expect it.

If nothing in writing: you can expect nothing because you are entitled to nothing.

Yes

You can quit

That or reach down deep into that ball sack and ask them what is going on.

Some places only allow for raises at fixed times of the year....

But ask the manager, or his boss about the situation, and why it was not automatic.

If they can't answer or otherwise explain, then look at your employment contract, and then get a lawyer involved.

Probably not unless you got it in writing

Taxes?