You must have agreed at some point - even if only verbally or in an email - that you are going to work and that the employer is going to pay you a certain amount.
That agreement - whether it's verbal or written - is your work contract. If some details (length of notice period etc) are not explicitly covered in it, then there are certain "default settings" which apply.
So to answer your questions:
1) Write down "Stundenlohn Fr. 24,50" (or whatever the hourly wage is)
2) Write down the date you started working, i.e. the first day that you did any work for this employer.
3) Write down the employer's name and address.
See? Not that hard.
However, there is no law as your employer seems to think which says "under 3 months, no contract required; over 3 months, contract required". Even if it's not in written form, you always
have a contract: just by working and getting paid for that work, you are considered to be in a contractual relationship. And there is legally no need for that contract to ever be made into a written one (although it's a pretty good idea, saves a lot of arguing later). You could work for the next thirty years on the basis of a verbal agreement and that would be perfectly legal.
So my concern is that the employer doesn't really seem to know (or perhaps, care to know) what he is talking about - which means he may not know (or care to know) some of the other details of employment law either.
In particular, you need to double-check that you're not accidentally working "under the table" - that your permit allows you to work, that necessary deductions (for taxes, unemployment insurance, pension contributions, etc.) are being taken out of your wages, and that things like accident insurance and employer pension contributions are being paid by your employer.