Is it legal to modify 2 existing logos into a new one? What's the difference between a copyright and a trademark?

Copyright protects an original work established in some sort of a permanent medium, such as a book, computer software, art, music, a play, etc. A trademark is a word or words, symbol, color, sign, sound, etc. that indicates the source of goods or services. Each logo must be unique and specific to the company. So you cannot copy parts of two other logos. However, you can create your own logo from common elements such a human figure, a star, a cube, etc., plus any word or combination of words allowed by the USPTO.

A copyright protects original works such as music and literature. A trademark is a source identifier used in commerce and protected under the Lanham Act. Trademark infringement occurs when another mark is likely to cause consumer confusion. It is not easy to determine if your combination of marks is confusingly similar to either of the source marks nor if your new mark might infringe upon other marks. This is a fact based question and you should check with an attorney to reduce your liability.

Do not combine two existing logos - as you will likely upset both owners of the originals. Be creative and make up your own logo design. Otherwise you could be at risk of a lawsuit from the original logo owners. Trademark and Copyright are different forms of intellectual property - and the laws that protect them are different. Trademarks are the "brand name" associated with goods. FORD (brand) cars; TIDE (brand) laundry soap; Budweiser (brand) beer; etc. A trademark distinguishes one party's goods from another party's goods. You come to associate the trademark (brand) with a certain level of quality for all goods bearing that brand name. Copyright is literally the right to "make copies" of the "works" protectable under the copyright laws. Original works protected by US Copyright Law are; (1) literary works; (2) musical works; (3) dramatic works; (4) pantomime & choreographic works; (5) pictorial, graphic and sculptural works; (6) motion pictures and audiovisual works; (7) sound recordings; and (8) architectural works. As you can see - either or both trademark and copyright protection may be available to the same item - each aspect of the law protecting a different aspect of the intellectual property.

You are probably all right to modify two existing logos to form a new and different logo as long as the new logo would not be mistaken by the consumer for either or both of the other two existing logos.

For generalized basic questions, e.g. difference between copyright and trademark, please see one of the various Nolo press books on the subject such as this one: http://www.nolo.com/products/patent-copyright-and-trademark-pctm.html. That will also help with your first question, although the answer will be fact specific. The oversimplified way of thinking about it is to decide whether the combination is distinctive and has no likelihood of confusion with the things it is derived from. However the combination cannot have components that are trademarks by themselves. Without more, an educated guess is that you likely cannot combine the elements without infringing, but if it is valuable to do so, you should consult a highly skilled trademark attorney.

That can be tricky. If the logos are fairly simple, then copyright probably will not come into play, otherwise, you might be creating a derivative work by making the new logo. As for problems with trademark, if your mark is confusingly similar to either of the marks you used to create the new one, then there is likely to be a problem. The best thing to do would be to talk with an attorney about the specifics and show the attorney your new design along with the sources of your inspiration.