There are a few red flags indeed. I am not from Austria, nor am I familiar with local laws. But a few things are universal.
Unless it is a hotel or AirBnb, never rent an apartment without seeing it. Putting aside the fact that those photos could be scrapped of a real estate web site or from AirBnb or simply stock photos, there are many factors that can make an apartment a nightmare: bad smells (like cigarette smell or dog pee), noisy neighbors, construction site next door (ok, that may be temporary), dogs that bark incessantly, something that is not visible in the photos because of the angle of the photo that is unacceptable to you, etc. Even when you see it, you may still miss something, but at least you've made an effort to eliminate apartments that obviously do not meet your criteria.
Never pay for anything before the contract is signed. Just reading a contract does not mean the contract is executed, it only means that you are getting informed about the deal you are about to enter into. It is not unusual for the first or first two months payment to be due at contract signing, which means that the contract is first signed and immediately after, a payment is made.
Always read the contract carefully. It may be in plain German, or in legalese, either way make an effort to read all the clauses. After you sign it, it will be too late to find out that you are responsible for all damages inside the apartment, including those caused by normal wear and tear. If, for example, the heating unit (if there is one in the apartment) dies because it is old, you do not want to be on the hook for replacing it.
Since you are planning to rent from a private person, you can negotiate contract terms. If you see something you don't like, say so, ask for it to be changed. If it is important and they refuse to change it, don't sign. Property management companies (companies that manage a large number of apartments) usually have standard contracts, and they are rarely negotiable. That does not mean you should not speak up, just in case they will agree to change the contract. If you see something you really do not like, do not sign.
Always make sure that the person saying they're the owner is indeed the owner. Even if he/she opens the door for you, that means nothing, as the apartment could be an AirBnb rented for a few days. Usually there is some governmental database that records the ownership of all real estate properties that you could consult. I believe somebody already mentioned a database that you can access for a small fee. That fee is worth it, even when the presumed owner is legit.
Always make sure you sign the contract with the owner or somebody that has a valid power of attorney. From the way you described the situation, it seems that the person you talked to intends to sign the contract, while the mother owns the apartment. That is OK if and only if that person has a power of attorney from the mother that covers the renting of the apartment. You would have to read that power of attorney to ascertain that is true. Some power of attorney comes very specific (X can manage the renting of my apartment), others are more general (X can manage my estate in Austria). If the person you talked to cannot or will not produce the power of attorney, stay away. There have been in the past people trying to take advantage of their relatives' property, and when the rightful owner realizes you are living in their apartment, they'll kick you out in a second, never mind whatever contract you think you have. If they do produce it, ask for a copy for your records. (I cannot say whether a simple copy or a notarized one would be required, it depends on local law.) The copy of the power of attorney and the contract together are what gives you the right to live in the apartment.
Sending money outside of the country may or may not be an issue. If there are laws regulating how rent is paid and if they require a local bank account, then sending the money to Italy is a clear red flag. Even without laws about the rent payments, it may indicate the owner's desire to do tax evasion. Income is taxed, but only if the fiscal authorities know about it. And again, it may be perfectly legitimate. On this one, I would look for advice from people familiar with Austrian laws and regulations. If you are unsure how to find out, the safer route is to insist on making the payments locally. (Bank accounts can be accessed from outside of the country. I have used my debit card when on vacation abroad, both at stores and at ATMs. All you need is to inform the bank you will use the card abroad, so they do not block it for suspicious activity, and possibly to pay some foreign transaction fee if the bank does not waive those.)