Much of this answer has already been covered in the comments, so this is pretty much just a rehash of what has already been said. It's not clear how much German you know so I'm going to try to answer at a very basic level.
First, unlike most European languages, German capitalizes all nouns, not just proper nouns. This includes occupation names no matter how they're spelled. But this hasn't always been standardized, and it may be possible that records from the 17th century use uncapitalized nouns. See What is the origin of the rules about the capitalization of the first letter of each noun? for more information on this topic. Anything written today would always use capitalized nouns; doing otherwise would be considered a spelling error. While German and English capitalization rules have other differences, this is most obvious one.
The mark above the 'a' in 'ä' is called an umlaut. Historically, this was a ligature combining 'a' and 'e', something like 'æ' in older forms of English. In German, the 'e' moved to the top of the letter and was reduced to the two dots as they exist today. There are only three letters with umlauts in German, 'ä', 'ö' and 'ü'. There are also capital versions of these, 'Ä', 'Ö' and 'Ü'. Note that the umlaut is not the same as the diaeresis which appears in some languages other than German; they appear to be the same but they have different uses and histories. There is one other German ligature, the eszet 'ß', which combines an old style long 's', 'ſ', with a 'z'. This should not be confused with the Greek 'β'. The eszet only exists in lower case, making a total of seven characters used in German but not in English. If the umlauts or eszet are not available for some reason, then it's customary to replace them with the corresponding letter combinations, so 'ae', 'oe', 'ue', 'Ae', 'Oe', 'Ue', 'sz'. In the past this might have been because the printer did not have them, or because a typewrite did not have the corresponding keys. With modern computer systems, it should be possible to reproduce these characters on non-German equipment, though figuring out how to do it can be a challenge. So while you may see Häusling spelled Haeusling, it should be avoided if possible. Other than that, they are just different spellings of the same word.
Umlauts rarely occur as the second of two vowels, so the Haüsling spelling you had in the original version of your question would be highly unlikely.
As to what a Häusling actually is or was, and how it could be translated into English, this is where I feel we're getting into areas of expertise outside the German language. Pre-industrial occupations were partly a matter of class and local custom, so there wouldn't necessarily be an exact correspondence between the occupations of different countries. Then there's the fact that some of these terms aren't in many people's active vocabulary, and that their meanings may have changed a good deal over time. I didn't know what a "cotteger" was until I looked it up, and it seems to have meant something different in the 17th century. So I think the best I can do is to offer my (with help from Google) translations of the definitions given in the link given above by Paul Frost. It defines Häusling as:
Small farmers with their own house and little property. See also Häusler. In some regions also cohabitors or renters of a small house without their own possessions. They mostly worked as day laborers or small businesses.
It defines Häusler thus:
Small farmers with their own house but little or no real estate and little or no livestock were called Häusler. The Häusler often only had employment opportunities as small craftsmen, servants, day labourers, schoolmasters or shepherds. In the 19th century, the Häusler were a transitional form of day laborers for the respective landlords and dependent on this additional income, since their own agricultural property was not sufficient to make a living. Due to their weak social position, they were disproportionately burdened with taxes and duties from the landlord in most communities.
