Is the contraction from that will to that’ll an actual word or not?
Well, that'll is not a word but a contraction.
Some dictionaries include it, some don't. That'll clearly exists, and is used to some degree. It's just a matter of whether it has been used enough to be widely understood.
An example of its usage would be in the song That'll be the day (1957) by Buddy Holly.
The Corpus of Contemporary American English has 3221 uses of this contraction. Since MetaEd mentions this as a part of the phrase "that'll be the day", I'll mention that the phrase appears only 40 times in COCA, so 99% of the time it is used in other contexts.
The Google n-gram viewer shows many uses back to the early 19th century: http://books.google.com/ngrams/graph?content=that%27ll&year_start=1800&year_end=2000&corpus=0&smoothing=3
According to Dictionary.com, that’ll is supported by the American Heritage Dictionary of Idioms as part of the phrase “that’ll be the day”, attested since the 1950s.
“That'll do”, a phrase used, among other things, to send your sheepdog home (and in that context made famous by the movie Babe), is found by Google Ngram Viewer in books and magazines since the early 1800s, including Hard Times by Charles Dickens.
- "that'll" is most likely not in any dictionary; it is an artifact of informal speech.
- it is commonly recognized and understood as a natural utterance. That is, the Queen of England might say it, but she probably wouldn't write it.
- you probably don't want to use it in a formal written document or in formal speech unless you are specifically trying to evoke informality.
- it is as much a 'word' as any other contraction (just not as common). Are contractions considered 'words' or are they all in this weird conceptual space between words and multi-word expressions?