That depends what they do with that money.
There's a concept in economics referred to as "the velocity of money", which is essentially a measure of the rate at which money circulates in a particular economy. Different groups can have a higher overall "velocity" than others. For example, the velocity of the income of a poor family is significantly higher than that of a wealthy one, because a larger proportion of the money they receive tends to be exchanged directly for goods and services.
To use your analogy, the rich man used their $100 to pay the poor man. This money simply went straight to the poor man. The poor man receives the $100 and uses it to pay their rent, which goes to their landlord; to pay for food, which goes to a local supermarket; to pay for a night out with friends, which goes to a local bar; to pay for a new computer, which goes to a PC manufacturer; and so on. And all of these transactions then filter through multiple more hands.
$100 in the hands of a wealthy man tends to have lower "velocity" than $100 in the hands of a poor man, because the poor man necessarily relies on that money being exchanged for goods and services more than a wealthy man, and after being exchanged for those goods and services it will then circulate to their employees, their supplier, etc.. The money tends circulate through the economy quicker once it's in the hands of the poor person than while it is in the hands of the wealthy person. This is kind of how wage-based economic models work. There are lots of examples of employers choosing to pay their workers higher wages and this ultimately resulting in a better overall economy, because the money circulates better.
They do. They are more likely to exchange the money for essential goods and services that then filter back into the economy than, say, to pay for an investment or to take the money abroad.
Actually, in economic terms, the value of someone with $100 who intends to spend it is very different to the value of $100 in the hands of someone who is either going to just hold on to it. The first person's money filters back into the economy, the second person's money doesn't.