Are there large discrepancies between what Chinese consider to be words?
"Q: Are there large discrepancies between what Chinese consider to be words?"


No. It is very clear what is a word and what not. A word can consist of one or more characters. Thus, a single character can be either a word by itself or is part of a polysyllabic word. Like in every other existing language, a word does officially either exist or not.

Q: Do even native Chinese generally split characters into words (词), or is the 
concept of words something we have forced into Chinese because it makes 
sense in western languages?


You can split a word into a character, but not a character into a word. If your question is, if Chinese was influenced by western languages to have polysyllabic words then the answer is "no". The reason why they do, is because there is not enough monosyllabic words to accurately describe every possible meaning. Nowadays words often consist of two or three characters (or even more). 

As an example:
While “书馆” can be a "teashop with performance by story tellers", "private school (in former times)" , or also "library (of classic texts)" it is clear that “图书馆” is a "library". Obviously, the more characters, the more accurate the word.

Q: Is a 词 different from a word in any sense?


“词” means word and can, as mentionned above, consist of one character or more.

Q: Do Chinese people generally agree upon how a sentence should be 
segmented into its compounds?


Yes. As mentionned above, there is clear grammar rules on how to construct a sentence. Nevertheless, people might have a different sense of choosing words to express their meanings. The result will be one less and one more beautiful sentence. But if a linguist checks both sentences, either of them can be grammatically correct or not.