Washington Post ran an article after Democratic Representative Anthony "don't let US government use the internet please!" Wiener resigned in 2011 following a sexting scandal. The article covered some details of this but not fully:
In the Senate, staffers who lose their boss keep their jobs for only up to 60 days as employees of the Secretary of the Senate — no matter whom they worked for or how long. “I was out of a job,” a staffer for the late Sen. Edward M. Kennedy told us. “The job that you had ends with the member.”
The minute a representative leaves office, the staff begins working for the district and answers to the Clerk of the House, Karen Haas. The nameplate comes off the door; the office phone is answered generically. (“Twenty-sixth District of New York,” in Lee’s case.) Staffers provide constituent services in the D.C. and district offices, but there’s no voting power. The arrangement lasts until the swearing-in of a new member — who then has power to retain or fire any of the old staffers. From an employee standpoint, the House is like 435 small businesses constantly facing takeovers, hostile or benign.
Texas Tribune covered a similar 2012 situation, though in this case the congressperson resigned to take another job. The article details offer some promising leads on who can be asked for more details (House Administration Committee):
State Rep. Veronica Gonzales, D-McAllen, will leave her seat June 30 when she officially resigns to take a position as the University of Texas-Pan American's vice president for university advancement. All of her staff members except one at the Capitol and one in her district office will be let go.
"Her last day is all the staff's last day as well," said Jennifer Saenz, who provides general counsel for Gonzales' office. "Then we're all unemployed."
The two remaining staff members will report to the House Administration Committee. Their duties will mostly consist of taking calls from constituents.
There are no official rules that outline how the office should be run between when the vacancy occurs and when the new representative is elected — that is at the discretion of the House Administration Committee — but Gonzales' situation is typical of how vacant offices are run, said Steven Adrian, executive director of House business operations.
As far as unemployment etc..., since the staffers are ultimately employed by the House of Representatives, the House employment policies kick in.
Also this NYT arricle gives a fluffy human interest story about poor staffers applying for unemployment in 1994. So presumably they can.
Also, Congressional Management Foundation has a handy publication called "Closing a Congressional Office".
It mentions unemployment explicitly, as well as offers further contact:
Advise employees to obtain current information on issues of unemployment insurance and retirement programs by contacting the House Finance Office and the Senate Disbursing Office