shouldn't (someone)(do something)?

The speaker asks "Shouldn't a woman have...?" She really means that "A woman should have..." People ask questions like this when they're trying to debate an issue and persuade people. For example:

Eighteen-year-olds are able to vote and serve in the military. Shouldn't they be allowed to drink as well?

Asking this type of question (which is called a rhetorical question) makes it seem

a (woman/man/person)

In the example above, "a woman" is used to talk about women in general, not just one specific woman. You can think of it as talking about an example woman.

Later in the sentence, the speaker says "her own life". The word "her" matches with "a woman". Both are singular.

have the right to (do something)

Americans have a strong belief in "rights". The Constitution of the U.S., which is the document that contains the country's most important laws, gives Americans several rights such as the right to carry weapons and the right to free speech.

So what does it mean to "have the right to" do something? It means that you are allowed to do it, and no one should be allowed to make you stop doing it. So the sentence:

Shouldn't a woman have the right to choose what she does with her own life?

...means that women should be allowed to choose whether to stay home or go to work.

Other "rights" that you may hear people talk about include:

• basic human rights (things like food, clean water, and not being made into a slave by anyone)
• civil rights (not being discriminated against or treated badly by the government)
• the right to remain silent (for someone who is arrested)

do (something) with (one's) life

The phrase "do ___ with your life" means to do something as part of your daily life, like as a career or as a hobby that you spend a lot of time on.