Is it wrong to call yourself Dr First_Name Last_Name when you have a Professional Doctorate such as DTech or DBA and not a PhD or MBBS? Is it wrong to call yourself Dr First_Name Last_Name when you have a Professional Doctorate such as DTech or DBA and not a PhD or MBBS?
This is highly dependant on legislation (of the country where you are when you are calling yourself). In germany (AFAIK, one of the more picky legislations wrt what you can call yourself):

Let's assume you move to Germany with a proper DTech from South Africa, Australia, UK or Sweden. Proper meaning: scientific thesis, degree from a university (or similar recognied highest level institution), honorary degrees have separate rules, bought degrees cannot be used in Germany. 


For South Africa, the most general rules apply: you are Firstname Lastname, DTech (University of ...)* you can also add a translation: (Doktor der Technologie).
If the country doesn't use the latin alphabet, you can transliterate. And long form or usual abbreviation are both fine. 

* Whether you put the degree before or after the name doesn't matter.
For Australia, there is a treaty on equivalency of degrees and in consequence (legal text in German) you are Dr. Firstname Lastname. Not Dr.-Ing., though.
Since Sweden is member of EU (same for EWR), with a Swedish DTech, you have the choice. You can either be Firstname Lastname, DTech or Dr. Firstname Lastname. Again, not Dr.-Ing. even though that is the German degree corresponding to the Swedish DTech.  

This does not apply to so-called Berufsdoktorate where the doctor is awarded not for a research thesis but automatically when finishing a Master level education.
UK used to be in the EU until recently and the offical texts have not yet caught up with Brexit. I saw a note somewhere that UK degrees are still treated like EU.




German doctorate degrees are mostly of the form Dr. ... ..., e.g. I'm Dr. rer. nat. (doctor in natural things [sciences], in my case chemistry). 


When addressing a person, usually at most the "doctor" is pronounced, the type/field only if that is considered important for the specific occasion.
I could ask to have the doctor put into my ID card where it would appear as DR (the line is all uppercase) only, without specifying what kind of doctor I am. 
I could also put the doctor to my doorbell, either as Dr. rer. nat. C. Beleites (sounds very arrogant, unless it is part of the professional doorbell Chemometric Consulting Dr. rer. nat. C. Beleites where I'd think it only a bit stuffy -  as it is, the office door bell actually reads Chemometric Consulting C. Beleites) or as Dr. C. Beleites (slightly arrogant). Only my business card contains in small print the information that I'm Dr. rer. nat. and Dipl.-Chem. (since one cannot know from Dr. rer. nat. that I'm chemist).





  BAppSci (Chiropractor), BAppSci (Osteopathy) that call themselves Dr.


This is not allowed in Germany. Even a physician can call themselves Dr. only if they actually obtained a doctor degree. Otherwise, they are
Firstname Lastname, Arzt (physician, meaning they passed their state exams and got approbated) or Firstname Lastname, Facharzt für ... (meaning they also passed the specialization exam for field ...).

If they also did their doctorate, they are Dr. med. Firstname Lastname, Facharzt für ....




  whether having a DTech allows one to formally/officially to be designated as a Dr. P.S.


Not in Germany. 

In some faculties, the candidate can ask to get e.g. a Ph. D. rather than the usual Dr. rer. nat.. But once that degree is awarded, one has to stick with it.