Well, several things to rule out first:

Are you on septic or city sewer?

If on city sewer, are your neighbors getting flooded also? Combined sewers (storm and sanitary) can cause the problem you are describing.

Also, do you have any downspouts that connect into drain lines that go to your sewer? They are usually clay tiles that the downspouts empty into.

Let us know.

It sounds like the sanitary and storm systems are connected if if "sewage" is coming up through a drain in the floor. I'm surprised that you don't have a constant sewer smell. It sounds like you are going to have to investigate and get the two separated.

We are on the city sewer system, not septic tank.
I will check with my neighbors asap.
There are no downspouts from the roof except in back; the sewer goes from the front of the house to the street.
This is a new problem apparently. The previous owner said they never had any problem, and I believe him.

To follow up on my adventure with the sewer: I found that the sewer had backed up again after the last visit from the plumber. I called Mr Rooter, a local franchise of a national company that does free Television inspections of the sewer line. The other plumber wanted $300 to bring in a camera inspector. I watched the video and it was so murky we couldn't see any break. They decided to jet it out with water. The hose got stuck, they determined in a break in the line. They dug, way down, almost 7 feet. They found not a break but a huge gap of 2 feet or more. And the cast iron line going into the house was rotted out. So, mystery solved: rain water pouring off the roof (I have no gutters in front) was pouring down into this corner of the house, pushing sand and mud into the pipe and water.
Mr Rooter provides a "trenchless" pipe liner, an innovation from Europe that involves resin epoxy and a plastic sleeve. It was quite an operation, taking all day and well into the night. They will have spent 2 and a half days on the project before it is done. Good news is I will have a virtually new pipe inside the old one with no damage to the yard (except where they had to dig). They will install a new clean out, a back flow valve, and all this for a mere ... BOHIC (bend over here it comes) $6,000. It is guaranteed for 50 years. I hope to never have to a sewer problem again, not in this house anyway.

Here is a link that might be useful: Mr. Rooter

Guarantees are only as good as the lifespan of the company behind them.

Six grand for less than a day's work???? Hmmmmm. No profit there.

Thanks. What a guy! Just what I needed after all this. I'll hope the company stays in business. The owner is a young guy who is settled here and if he serves other customers the way he did me, he'll do well. The alternative was to patch the line or replace by digging the entire thing up. The former would have cost me 2-3k. I'm guessing that digging and laying a new line would be about 7 grand plus repairing a 50 foot scar to the yard. I doubt I could have gotten a new line for less than I'm paying them.

Let's face it, plumbers have us where they want us with sewage backing up into the basement, a man is not in a position to bargain very effectively. By the way, the work has taken them most three days, going on four now, and it involves a lot of expensive equipment and material. I really hope they are making a good profit; they deserve it. The two guys were here until midnight last night monitoring the line while the resin set. And they were here early this morning in pouring rain to finish up. I'm guessing there are 70 man hours in the job already and another 8 to 12 to come. I really haven't worried about that. I'm just hoping the problem is solved and I'll never have to worrry about it again.