Can a company send you an offer letter with your salary, then withdraw the offer after you resigned from your current job?

You may be able to sue. This might be breach of contract (it depends on exactly what was said in the offer letter); you might also have a cause of action under the theory of "promissory estoppel." When party A (the prospective employer) makes a promise to party B (you) with the intention that you rely and act on it, and while knowing (or at least, they should know) that to rely on it, you will have to do something significant to your detriment (like resigning from an existing job), that can be enough to make the promise enforceable.

Under either theory, you would most likely be entitled to several months of salary if you sue and win; the exact amount is impossible to state in the abstract, since much depends on how long it would reasonably take for someeone at your level, with your experience, in your field and geographic area, to find a comparable job. You would seek and might be awarded an amount of pay commensurate with how long you might reasonably be out of work, since that represents the damage you have suffered.

It's also impossible to say what the lawsuit might cost. The good news is, you probably don't need much in the way of expert witnesess--perhaps a recruiter or career coach or two, to testify as to the impact on your employment. That will help keep costs down. The main variable, though, are attorneys' fees, which can vary wildly--for example, in my own building, there are lawyers who charge from $125 to $250 per hour; when I had my own employment case and hired an attorney to help me (it's unwise to represent yourself), he charged me $400 per hour. So what you'll pay depends on who you retain as counsel.