As for real estate: of course you can own it but there are very strict rules as to what you can do with it. It needs to be rented out full time etc pp. It is not to be used at your leisure. You are under no circumstances allowed to live in it, regularly use it whatsoever. Also they will tax you for its income.
Same goes for rental apartments or the apartment your mother rents for you and you have the key. Famous examples would be that most famour German tennis player, supermodel, several pilot court rulings that even just shared one apartment between multiple pilots at their home base airport etc.
Shares in a company: Income
Tax residency has multiple levels. You can be 100% of the
income tax hook, you can be off it after some time, you can be off it for a certain percentage. Depending on where you move that partial taxation goes to extreme levels and is tied to timeframes - most notorious example would be "Ueberdachende Besteuerung" if you move to Switzerland from
Germany. Special clauses just for that country. Owning shares in a registered company puts you in that "partial taxation - dont know english wording for this" tax code which will put you in a lot more stricter situation in general, adds another "indicator" and in general puts you back into the income tax system which creates new problems.
As for hair dresser etc, just the hair dresser is not going to trigger
tax residency, visiting the same doctor over and over again can (court ruling). Again indicators.
You can have kids just not a wife. Kids need to be registered with the mother and you must not be married or have an equivalent relationship.
Nobody needs to proof your 120 days. You will get a letter from the Finanzamt and will need to proof that you stayed in another / your tax residency country longer than you did in Germany.
All of the above points are flexible (except access to apartment / real estate, married etc) and yes this is very basic and more complicated in general. A knowledgeable tax attorney can explain you in detail. Make sure to ask one that actually deals with this on a daily basis. Regular local tax attorneys usually dont know s**t and will only cite the basic 183+, center of life blah rules without those specific examples, court rulings.
What they do is use them as "indicators". If you have fulfilled multiple indicators or just one hard fact you will be treated as income tax resident, having commited tax fraud etc. It will be up to you to proof otherwise not them. It will be up to the judge to interpret the pretty much non existent concrete wording for all of this and accept prior rulings or not.
This is all Germany but very similar in most developed EU countries. This is not "overakill points" but from court rulings or plain tax law.