The Mehlhorn family name was quite common in Saxony at that time. You can find many (older) records of other researchers at
http://gedbas.genealogy.net/. For more historical details of Lößnitz see
http://hov.isgv.de/L%C3%B6%C3%9Fnitz_(1)
I have checked the Lößnitz address directory of 1926 and found 33 entries for that name, including "your" Paul Max Mehlhorn:
http://digital.slub-dresden.de/werkansicht/dlf/105292/535/0/ . His occupation was: Schuhwarenzuschneider. Means he prepared and cut material (leather) for making shoes. Before the industrialization, shoes were produced often at private family workshops in the homes of the people. It seems he was not officially a shoemaker, but rather a supplier.
As the family you are looking for is rather young, it will be difficult to find a lot of information online. You could contact the Lößnitz parish and ask if Paul Max can be found in their church records.
Contact: Evangelisch-Lutherische Kirchgemeinde Lößnitz-Affalter
Rathausplatz 3, 08294 Lößnitz
Phone: +49 3771 - 555110
eMail: pfarramt@kirche-loessnitz.de
http://www.kirche-loessnitz.de/
However, starting from 1875 all births, deaths and weddings in Germany also had to be registered at the civil registry office (Standesamt), These offices can provide the corresponding records, but documents are usually only made available to direct descendants or spouses of the person concerned.
At
http://list.genealogy.net/mm/listinfo/sachsen-l there is a genealogy mailing list for Saxony. I could try and post a request there. Please provide any additional information and sources that you have for that specific family.
Is there already a WikiTree profile for the person(s) you are asking for?