Interior Solution for profit maximisation problem A function $c: \mathbb{R}^K_+ \xrightarrow{} \mathbb{R}_+$ is is said to be a cost function if

The value of function $c$ at $y = \textbf{0}$ is $0$: $c(\textbf{0}) = 0$
$c$ is continuous on the domain $\mathbb{R}^K_+$, strictly increasing and strictly convex on $\mathbb{R}^K_{+}$
For any $p \in \mathbb{R}^K_{++}$, the set $B(p) = \{y \in \mathbb{R}^K_+| c(y) \leq p \cdot y\}$ is compact, the interior of $B(p)$ is nonempty.

Suppose a firm has cost function $c(\cdot)$ and $p\in \mathbb{R}^K_{++}$ is the market price. What sort of topological assumption I need to add to guarantee the existence of an interior solution to the profit maximisation problem?
$$
\pi (p) = \max_{y\in\mathbb{R}^K_+} \left\{ p\cdot y - c(y) \right\}
$$
Remark 1: From assumption (2) and (3), the existence of solution is guaranteed. Due to strict convexity, uniqueness is also guaranteed. However, the solution is probably not inside the interior of $\mathbb{R}^K_{+}$. I could not find an example for this at the moment.
Remark 2: My prototype cost function to look at is $c(y) = y_1^2+y_2^2$. However, the cost is not necessarily to be additive separable.
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.