1) It is not entirely true what you wrote about linear equations. If your equations are contradictory, then no matter how many additional variables you add, you will not find a solution. For instance $x+y=0$, $x+2y=1$, $2x+y=1$ has no solution regardless possible additional variables.

2) If you mean *homogenous* equations, that is, the constant terms are zero, then your claim is right. Well, it is obviously right since taking all variables to be zero gives you a solution. However, it is true that in this case if the number of variables is larger than the number of equations, then you have a *non-trivial* solution.

3) For homogenous systems of equations over $\mathbb C$ the same bound holds. Here is why: since it is  homogenous, again, taking all variables to be zero gives a solution. Therefore the hypersurfaces determined by the equations have a common point and by the [affine dimension theorem][1] (which is essentially saying that one equation can cut down the dimension of the solution set by at most one) the solution set has positive dimension. 

4) Over $\mathbb R$ you run into the problem of not having solutions no matter what. If your equations contain something like $x^2+y^2+...=0$ involving all the variables, then you are out of luck. 


  [1]: http://books.google.com/books?id=3rtX9t-nnvwC&pg=PA48&lpg=PA48&dq=affine+dimension+theorem&source=bl&ots=XO-81JMIj7&sig=h9Enqrc9HGpdGH5XSLG7h7CXxnk&hl=en&ei=XimCTcjgFYG6sAPal53rAQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q=affine%2520dimension%2520theorem&f=false