How can I make the variance of a multiple sum of set of fixed number of variables minimum?
If you're willing to replace 'variance' with 'max minus min' (spread), then you could formulate this as an instance of integer linear programming and apply an off-the-shelf ILP solver.  There is no guarantee that it finds an optimal solution in a reasonable amount of time, but if you're lucky, it might complete without taking too long; or if you stop it early, you might get a decent solution.
To formulate this as an instance of ILP, add zero-or-one variables $x_{i,j}$, with the intended meaning that $x_{i,j}=1$ means that person $i$ is assigned to team $j$.  Each requirement in the problem statement can be expressed as a linear inequality; for instance, the requirement that everyone in a team has a different can be expressed by inequalities of the form $x_{i,j} + x_{i',j} \le 1$ for each pair $i,i'$ of people from the same seed.
To minimize the spread, introduce variables $\ell,u$.  Let $v_i$ denote the value of person $i$ (this is a constant, not a variable).  Add linear inequalities $\ell \le \sum_i x_{i,j} v_i \le u$ for each team $j$.  Finally, minimize $u-\ell$, subject to the inequalities above.
If you want to use the variance, this becomes a mixed-integer quadratic program instead of a mixed-integer linear program, and I think quadratic programs are significantly harder.