How is y=mx+b different from hθ(x)=θ0+θ1x?
1) Your hypothese $hθ(x)$ is clearly a linear model with $b \leftrightarrow θ_0$ and $m \leftrightarrow θ_1$ as you expected (don't be too hesitant and your colleges could have been able to confirm this).

2) Lets call $hθ(x_i)$ the prediction $p_i$. A cost function $C$ can be any function that when you try to minimize it holds a good solution.
One example would be the mean squared error (mse):  $C(p_i, y_i)=1/2 \cdot (p_i - y_i)^2$. I'm quite sure the mse is used to optimize the linear model, however other cost functions are possible.

3) I suppose this is an assignment? You clearly gave a very good suggestion to approximate your data. I suppose the goal of your task was to come to the solution of the linear model yourself. Try to understand how $b$ and $m$ is estimated in the linear model.

If you are in school, it is very useful to understand how people before you came to certain formula. It seems like you are reinventing the wheel, but the insights you gather with this can only help your education :).

Good luck.