As hexeliebe correctly points out, the invention is a work product, so it belongs to the company. You are right in that a patent is granted to a person or persons -- the inventors -- but the rights under the patent are typically automatically assigned to the company.
What are my rights? If I refuse to sign, the company will just take my name out, can they do that?
You really don't have any rights -- anything you invent on company time gets assigned to the company, period. When you assign the rights, you assign ALL of the rights, so you have nothing left except the knowledge that there is a patent in your name. You can order a plaque to hang up in your office if you want (I did that with mine...).
If you refuse to sign, the company can file to have your name removed from the patent. It's not cheap for the company to do this -- they have to pay a lawyer or patent agent to file a petition and pay a fee -- but it's cheaper than dealing with an uncooperative inventor. As long as the company makes a good faith effort to get you to sign, and then files a petition and fee, they can remove an inventor's name from the patent.
Simply put, you have no rights to the invention, and no leverage to get the company to pay you or do anything other than they are already doing.
Now, I do not want to do that - because it will mean that I will not be able to use this idea again if I leave, right? So - when I leave, they can apply for the patent without me?
See both things above. First off, if you invented something on company time, it belongs to the company, not you, even if you leave the company before the company files for the invention.
If you are difficult, the company can use the same petition to remove your name as above.
Remember, an invention consists of two steps -- conception, then reduction to practice. If you conceive an invention, and someone else reduces the invention to practice (builds it, for example), then you are both inventors. If you leave the company and become difficult, then presto -- they'll remove your name and the other guy will be sole inventor.
I'm not sure why you are being so stubborn. Just sign the paperwork. Let them correct the errors -- its not like you'll be held liable or anything if the errors don't get corrected. If the patent eventually issues, you can always call yourself an inventor...