Interior Solution for profit maximisation problem
Your assumption 3 is compatible with corner solutions of the kind $y_i=0$ for some $i$, and is not sufficient to avoid corner solutions for some prices small enough.
With production functions it is common to assume Inada's conditions to avoid corner solutions. With cost functions, such conditions are quite naturally expressed as
$$ \lim_{y_i\rightarrow 0} \frac{\partial c}{\partial y_i}(y_i,y_{-i})=0 $$
$$ \lim_{y_i\rightarrow +\infty} \frac{\partial c}{\partial y_i}(y_i,y_{-i})=+\infty. $$
In your discussion, you proposed the cost function $$c(y_1,y_2)=y_1^2+y_2^2+2y_1$$ to illustrate a corner solution. This function does not satisfy the first Inada conditions wrt $y_1$, hence the corner solution.
A general comment to conclude: it is not fully natural to forbid firm's exit from a given output market. In IO there is a quite dynamic literature on entry and exit. And no firm is producing all products available in the economy.