Possibly relocating to Singapore...so many questions.

Sorry to say, only spouse and legal child or legally adopted child by legally married couple would be opted for dependent passes. So, you and your child are not entitled for dependent pass nor (Long Term Visiting Pass) LTVP. Below Govt links for your reference:

http://www.mom.gov.sg/passes-and-permit … /key-factshttp://www.mom.gov.sg/passes-and-permit … ligibility

So, you and child can visit here as a tourist for 30 days (which can be extended max 89 days which is hard to get an extension as you and your child do not have legally relationship with the Employment Pass holder). I don't think other questions are relevant to response. You are coming into a place which has very strict rules or in another word "a different set of rules" in place, which you may not even heard off. Good luck.

What Surya wrote above is not entirely correct:
By the letter of the law, the Singapore authorities accept legally registered partnerships from other countries. So, if you have a German "eingetragene Lebenspartnerschaft" (and your partner fulfills the other requirements), you can get a DP. But don't expect this to be easy!
Also, you need a work visa (i.e. your own EP or similar) to do any work in Singapore - but there is no such visa possible for self-employment in a foreign-registered business. Even if the chances of being found are low, Singapore is NOT the place to break rules!

Regarding your other questions:
- Full day child care in a local facility cost S$600-1200/month, international kindergartens more.
- A live-in maid gets S$400-600/month plus room and board. With insurance, maid levy and all other costs it comes to S$1200-1500/month.
- There are no comprehemsive health insurances in Singapore as what you know from Europe. Most people have none or just accident and hospitalization cover. For more, you need to buy an international expat insurance.
The health insurances that companies offer to their employees differ widely in scope, so do read the fineprint! In many cases family members are not covered.

Beppi: My above statement was into consideration that they are not legally married or not recognised legally as couple. So, when someone providing legally registered partners letter or certificate from Germany or any other country, means they are recognised by the local govt with their living status as couple and child as legally child of them. In that case my above statement is not applicable as they are already recognised as legally couple or partners.
In addition, the EP holder must adhere to fulfill the day to day expenses and other requirements such as insurance coverage which has to be proved before MoM will consider to approve dependent passes for the legal partner and their child.

Few other points, full day child care may starts with $800 to $3000 (international kindergarten can fetch more than $2k to $3k). A domestic helper minimum salary is around $550 ( may get Myanmar or Indian at lower side of $500) and can go up to $1000. Levy charge starts with $265 per month. Onetime helper insurance around $500.

From personal experience I know that if the primary person has a work permit (I.e. EP not Spass or other such) and can show a common law marriage status, they and their child can get a visa. I had a friend from England in the same situation and they just needed the British embassy to endorse their status, which was pretty easy and it was accepted by MoM. This was however 10 years ago so not sure if the rules have changed since then.