A name like Nike

A logo like the swoosh logo

A slogan like “Just do it”

A song, book, film, or other original creative work. Get a copyright instead.

An invention is like a technical process, the machine, manufactured good, or chemical formula. Look into a patent.

Frequently no, because it could confuse.
Though the other mark is employed for different products or services, it might be viable because consumers may not be confused about who sells what.
Take the Delta brand. Delta Dental, Delta Air Lines, and Delta Faucet Company coexist because the products are different enough to prevent consumer confusion.

Yes.
One trademark—that’s one name, one logo, or one slogan—per application.
You can register a logo that contains text, but then the structure and the text together are considered one trademark.
Want them covered individually? You’d need to file two applications.

It depends on what you sell.

The USPTO(U.S. Patent and Trademark Office (USPTO)) puts all products and services into 45 classes. 

Possess a restaurant? You provide service in class 43 (food services). Also, sell cookbooks and teach cooking at the restaurant? You may want to register in class 16 (paper goods) and class 41 (education and entertainment services) too.

Usually, your trademark security is limited to the classes listed in your application.

The USPTO(U.S. Patent and Trademark Office) discussing attorney might reject your mark if there is a chance consumers could mistake it with another trademark (i.e., if your signature sounds like or looks like another mark in a similar industry).