There are minimum age restrictions.

As long as s/he is capable of signing a legal contract (usually at age 18, I think).

There are, or at least used to be, medical restrictions, i.e. must not have venereal disease.

I don't see why the government needs to "protect" me from this. As far as I'm concerned, this is another example of government being too involved in the bedroom.

There are consanguinity restrictions, e.g. double first cousins cannot marry.

Marry? I don't see why not. Children? The risk is high (depending upon direct blood relationship) for genetic defects. Is it the government's job to prevent genetic defects?

Does the government take an interest in the reproductive habits of other carriers of "bad" genetic code?

Clearing the past religious clutter from the contract would be a great idea. You could, in advance, assign financial responsibility for any children conceived during the contract period and so on. You could even do away with "prenuptial" contracts. What would be the purpose of a "pre-contract" contract? If either party has specific pre-existing resources that they do not wish to have included for consideration in the contract, then they can include the required verbiage in the "marriage" contract.

Requiring monogamy is possibly a religious restriction.

True. I don't see any reason why multiple husbands/wives (even multiples of both, one guy marries 3 wives, and two of those wives have other husbands) couldn't be handled under contract law. It would make for some VERY interesting cases in the event of a divorce.