I am a California attorney and not eligible to give legal advice in your state. The following is information only, based on federal law and general legal principles. YOUR STATE MAY HAVE ITS OWN LAWS THAT PROVIDE SIMILAR OR GREATER PROTECTION.
To respond to your question, court records are public records, which means anyone can see them. Many courts these days have databases in which a researcher can search for parties by name. When names are similar, the researcher won't know for sure who the person is, but other information will often reveal the identify further. For example, if you sued your former employer and also listed that same employer on your resume or work history, the match is pretty easy to make. Other information, such as city, apparent age, sex, race, whatever can be used to narrow down who the party is.
If you want to hide the fact that you sued, it will be pretty difficult given the ease with which a person can conduct identify searches. If you omit a former employer from your resume or work history, most potential employers will note a gap and either ask you about that or pass you by in favor of someone who holds no mystery.
Best is to be upfront about it, and explain why you took that route, such as you brought the issue up to the employer numerous times with no result.
If prior litigation is interfering with your employment opportunities, you may wish to retain an experienced plaintiffs employment attorney for a consultation to help strategize a way to deal with this.
To find a plaintiffs employment attorney in your area, please go to the web site of the National Employment Lawyers Association (NELA). NELA is the largest and most influential bar association in the country for attorneys who represent working people. The web site is www.nela.org, and you can search for attorneys by location and practice area.
Also, NELA has affiliates in every state and in many cities. On the NELA web site, you can look at the list of affiliates. Some attorneys will be listed in the affiliate membership list, some in the national organization membership list, and some in both. Being listed in one or both lists should not affect your selection because listing in the national on-line directory is a marketing decision.
It depends. If you filed a claim with the EEOC or the FL Dept of Human Rights and the proceedings were confidential, then they will not be able to find out about it. If, however you actually filed a lawsuit in court, that is a public record and your prospective employer will be able to find it, assuming that they do a public records (i.e. search the clerk's records) search. If this comes up you should be prepared to explain what the suit was about, why you sued, and re-assure your prospective employer that you won't sue them for the same reasons because after all they are a much better outfit than your former employer, right? Good luck!