Is there a word or term used to categorize words that are unnecessary to conform with grammar rules and don't add to the meaning of a sentence?

A pleonasm is "the use of more words than are necessary to express an idea" (here and here)

You gave examples involving the sentential complementizer 'that'. When occurring after a propositional attitude verb like 'claims' and 'believes', the word is entirely optional. But it is a matter of style and preference whether or not you should get rid of it. Just because it is not necessary doesn't mean you should get rid of it.

Perhaps superfluous

Unnecessary, especially through being more than enough: the purchaser should avoid asking for superfluous information

Oxford Dictionaries Online

Consider the examples given in the American Heritage entry for superfluous

Synonyms: superfluous, excess, extra, supernumerary, surplus These adjectives mean being more than is needed, desired, required, or appropriate: delete superfluous words; trying to lose excess weight; found some extra change on the dresser; supernumerary ornamentation; distributed surplus food to the needy.

There are options.

• optional. This is the most likely word to describe the situation formally.
• facultative which means optional. This is a scientific term, used in many sciences, in opposition to obligate which means necessary. It is not used much in linguistics, but works.
• pleonastic is by definition, the use of more words than are necessary. This is most commonly used for content words like 'free gift' or 'repeat over and over'. It can be used to describe syntactic items but that doesn't say if one is allowed grammatically to remove the item. For example, in 'I am green', 'I' and 'am' are pleonastic, but neither can be removed.

So 'optional' is the most appropriate.

It appears to me "that" you are mistaken. That is acting as a subordinating conjunction. These are two simple sentences. He Claims. (what does he claim?) He was at the library yesterday. "that" is absolutely necessary to tie the two sentences together, unless you speak broken english! http://forum.wordreference.com/threads/that-subordinating-conjunction-or-relative-pronoun.1337388/ goody, goody gumdrops!