Why they matter
The Prophet, peace be upon him, said the conditions most deserving to be fulfilled are those by which you make intercourse lawful. Conditions in a marriage contract are not a modern imposition, they are part of how the contract has always worked.
A condition written down is a condition that can be honoured. One that was assumed is one that gets renegotiated the moment it is inconvenient.
The two that matter most
Where you will live. Joint household or separate accommodation is the condition most likely to be assumed by one side and denied by the other, and the one hardest to renegotiate after the nikah.
Continuing study or work. If she intends to finish a degree or keep working, that has to be agreed in advance, in specific terms, including what happens if there are children.
On the mahr
The mahr is her right and hers alone. Whether it is paid immediately or deferred should be explicit and written, because an unwritten deferral has a way of becoming a forgotten one.
Dowry demanded from the bride's family is a separate thing entirely and has no basis. Saying so in your stated conditions protects both families from being negotiated around.
Where the scholars differ
Some conditions are agreed among the scholars, and some are genuinely disputed, including conditions restricting a second marriage. Effect and validity are not the same question.
Ask a scholar you actually learn from before relying on a disputed condition. Do not rely on a platform, including this one, to settle it for you.
General guidance, not a fatwa. Where something is disputed among the scholars we have said so, and you should ask someone you actually learn from rather than relying on a website, including this one.