Yes, I worried about how this problem will affect my business when I apply for a trademark.
Remember, you are not applying for a trademark. Trademarks are created by USE. Registering them with the PTO gets you wider coverage and easier enforceability for your rights, but the rights themselves are created by you using the mark, not by the registration process. And even if you do an ITU application, you don't get registration till you use the mark in (interstate) commerce.
I also do not like someone else having ownership of my intellectual property. What problems will that create for me?
Mere ownership of a domain name does not translate into ownership of trademark rights in the domain name or any part of it. Trademark rights depend on the trademark being used as a trademark, i.e. as a brand.
To give some background on how trademarks work:
Say you establish a trade name for yourself, Flibbity Widgets, Inc. You use it on your letterhead and in your contracts. Do you have a trademark in the business name? Not yet. You need to use it as a brand for your goods/services: advertise the goods/services with it, mark it on the goods or their containers, etc. (While many businesses use their trade names as their trademarks, the two are distinct and it's possible to have a trade name which is not used as a trademark.)
Now, say someone gets you a domain name: Flibbity.com. As I understand it, even if that someone "owns" that domain name, there is nothing to prevent you from going out and establishing the domain names "Flibbitywidgets.com" or "MyFlibbity.com," etc. And (active practitioners, please jump in to correct or expand on this as necessary) the only way the owner of Flibbity.com could prevent another from using "flibbity" in their domain name would be through the trademark laws. But to do this, they would need to be using "Flibbity" AS A TRADEMARK -- not just owning the domain name.
My concern is my business name is the same as the former website/social sites. Would it be good idea to pursue this legally first. If not resolved how would this affect me when I apply for a trademark? Should I change my business name to my new url?
As I see it, the concern isn't how this will affect registering the trademark. The concern is what practical impact it may have on your business.
The above is nice ivory-tower legal theory over whether a guy sitting on a domain name has standing to move against you in trademark. When it comes to practicalities, however, more needs to be considered. For instance:
-If you don't resolve the contract issue, will the other party take you to court on the contract if you don't capitulate?
-If he breached the agreement in bad faith and knows it, would he readily capitulate to you if you if he heard from your attorney?
-Will he think he has more rights in the name than he does and try to put you in a defensive position either with regard to domain name or to trademark registration -- so that you'll have to hire someone to straighten him out?
-Would the lack of ability to use the name as you wanted impact your marketing?
If you can at all afford it, you should be consulting with your own counsel about this. Ideally, a trademark attorney who both knows internet law and has experience with contract disputes.
It is ALWAYS good practice to see a trademark attorney before investing in the name and branding of a new business, anyway. Some names make stronger trademarks (and so are more easily enforced) than others. Also, many people don't do a proper search of other marks that are out there, so that AFTER they've sunk money into promotion, establishing websites, tooling up with the name on the product, etc., they find themselves sued by someone else in a similar line of business with a claim to priority. The standard for trademark infringement is "likelihood of confusion". Unlike domain names and trade names, which can coexist if they're not exactly the same as another, trademarks depend on a multitude of factors involving such things as look, sight, sound, and meaning of the words as well as the type of products/services at issue and whether they are high end or low end, retail or wholesale, niche or general appeal, etc. And marks do NOT have to be federally registered to cause problems. Trademark searching is an art.
If you can't afford an attorney, certainly read up on trademarks -- what they are and what they aren't, what's strong and what's weak. The PTO website tends to address only the aspects that concern registration, but there's much more to it. This forum can be a good source of information, also the Nolo Press books on law for the layman are usually pretty well done and are available in many libraries. And there are a number of law firms that have explanatory articles for prospective clients that explain some of the basics. Searching a combination like "likelihood of confusion" along with "strong mark" or "strong trademark", "generic" and "descriptive", along with "common law" should bring up relevant articles.