Well, if keeping people happy is only a minor concern, then we have some solid options. I don't believe that such a situation will persist for long, as people are, by definition, very emotionally invested about their relationships and children. Also, your requirement for "keeping equal rights" will be almost impossible. That said, onward!
Discourage partnerships to begin with
If we assume that the government can "take care" of any number of children to any reasonable standard, people will be incentivized to produce children out of wedlock to begin with. This is really the optimal case from the legal perspective; one parent is a custodial parent and receives a government income, and the other is non-custodial, and pays taxes on their work income. In order to accommodate this:
Marriage is disadvantageous
To begin with, marriage will be heavily taxed. Government support will revolve around single parents; birthing classes will not accommodate non-birthers, parenting courses will avoid talking about two-parent situations, and schools, hospitals, and so forth will assume single parenthood on legal forms. Importantly, parents will receive the "baby bonus", which is taxable, whether or not they live with the child; this creates equality between the parents, and makes sure it is creating children that is incentivized, not child care. With the promise of comfortable government support, and no requirement for child support from a non-custodial parent, separation (or, more likely not moving in together at all), will be extremely attractive to both parties. Since separated parents should owe each other nothing, no kind of partnership should be officially recognized to do so. A limited list of reasonable major joint assets, such as a home or a vehicle, may be registered with the government for liquidation and division afterwards, but generally, joint ownership of anything is illegal. This means a custodial parent will likely loose out big-time on separation, and is really better off staying on their own to begin with. Since child rearing isn't really incentivized or desirable:
State child care should be the norm
Most people will want to have a state institution do the child rearing, and visit their child on an ongoing basis. This allows them both to maintain their careers, as well as equalizes parental time and standards of living for the children. To really encourage this, either parent should be able to veto in-home care in favour of state institutional care, right from birth. This will also keep people from abusing their children to achieve better marks to earn higher baby bonuses.
TL;DR
Make legal partnerships or marriages legally and socially difficult to begin, severely limit or eliminate joint asset ownership and division thereof on separation, ensure both parents are paid for creating the child regardless of the parenting arrangement, and normalize the institutionalizing of children from birth to create truly equal parenting arrangements.
Personally, I think the criteria given are ludicrously unacheiveable.
- Baby farming will be a popular choice of job. For parental (and
indirectly, gender) equality to be fulfilled, payment must be equal
to both parents regardless of their involvement in child rearing; but
this means that non-custodial parents will earn comfortable livings
for no ongoing effort, and your workforce will be hamstrung. If
non-custodial parents don't receive a payment, then a) your idea
about encouraging fair short-term or polygamous relationships will be
moot b) one gender or the other will likely be heavily disadvantaged
in proportion to the size of the baby bonus (just like now!), and c)
separation will continue to be complex and acrimonious, resulting in
enrichment of legal professionals.
- Baby farming requires no
education, but can be a comfortable living under this system (as it
indeed is to some degree now). The number of people who choose to do
this will depend on how competitive this income is, relative to other
jobs. If the payments are high enough, your workforce will be
severely disabled, but if they're too low relative to other jobs, you
create a disadvantaged population. This is historically not a
solveable problem.
- Lack of child support means that unscrupulous men
are incentivized to create babies and abandon them, placing an
inordinate load on the government system, and creating enormous
inequality for women. For the wealthy, men will especially be able to
lure women into becoming pregnant and then abandoning them as a form
of tax write-off. The ability of men to do this at a much higher rate
than women is an inequality that you will never solve without
requiring ongoing child support.
- The next generation will have the
same choices as their parents, but they'll be loaded by losing a
portion of their income to their parents. In order to insure that
children are properly cared for, baby bonuses will have to increase
to accommodate the loss of this income for the child.
- All the legal
incentives in the world won't really help with divorce much, since
division of money isn't really the worst part of separation, it's
the division of the kids. People are super emotional about it, and
probably won't respond in rational ways to legislation in the way you
think. This is somewhat mitigated by the above plan of minimizing
parental involvement at all, or at least restricting parental
involvement to a single parent; even so, children will be traumatized
when it happens, and your whole society will revolve around this
being very common.
- As a result of your plan to base compensation on a
child's education, teachers will gain enormous amounts of power over
people's lives, and the economy and politics of teaching will
drastically change. I won't venture to guess how this will turn out,
except to say that power has a distressing tendency to corrupt.
- Abortion will likely become more contentious, as some mothers will
want to bring babies to term regardless of the consequences, just for
the money. State control of abortion will be difficult to maintain.
