Can somebody who overstays a visa waiver adjust their status to obtain a work permit and residency?

Only if he marries a US citizen and does not get deported beforehand.

Your question requires an attorney consultation. It is not a simple question that can be answered on this type of forum. There are many factors that would need to be considered and evaluated. First of all, on what grounds would you be eligible for an immigrant visa and/or work authorization. I strongly suggest that you contact an experienced immigration attorney for a face-to-face consultation and give him/her all of the facts surrounding your situation. He/she would then be in a better position to analyze your case and advise you of your options.

You need to set a consultation to discuss the specifics of the situation. In many situations an overstay may not adjust or change status once out of status. Only in limited circumstances may someone do so.

Yes. But only through a marriage to a US citizen.

It is only possible if the person marries a US citizen. There is no other way to change status within the US for someone who enters on the visa waiver and certainly not if one overstays the 90 days. At that point, he/she starts to accumulate unlawful presence in the US and is subject to a 3 or 10 year bar to reentry if he/she leaves the US to change to any other status. The ONLY way to legalize one's status within the US after falling out of status is through a bona fide marriage to a US citizen.