Which insurance is compulsory in your situation?

In many cases yes. The obligation to take out health insurance is linked to being insured under the Long-Term Care Act, and anyone employed in the Netherlands usually is. European coordination rules determine which country has jurisdiction. With cross-border work, have this established in advance, because being insured twice or not at all costs money.

Not at present. The self-employed fall outside the employee insurance schemes and bear the income risk during illness themselves. Legislation is being prepared for compulsory basic disability cover for the self-employed; as long as that has not come into force, disability insurance (AOV) or joining a mutual support fund remains your own choice.

No, for nobody. A landlord can, however, require in the tenancy agreement that you have liability or glass cover, and VvE regulations can determine what is insured jointly. Those are contractual arrangements. If you fail to comply, that is a breach of contract towards the other party and not a breach of the law.

There is no compulsory employer's insurance. You do have an obligation to continue paying wages during illness and liability for the safety of your employees under Article 7:658 of the Dutch Civil Code. Many employers cover those risks with sickness absence insurance (verzuimverzekering) and employer's liability cover, but that is a choice, not an obligation.