Help trying to find Economic Profit for a Monopoly
Profit is revenue minus costs. So we have $q = 12500$ and $p = 35$. (Note that I'm assuming you've correctly figured this out.) Then we have revenue:

$$R = \int_{0}^{12500} MR dq = \int_{0}^{12500} (60 - 0.004q)dq$$

And costs:

$$C = 300000 + \int_{0}^{12500} 10dq = 300000 + 125000 = 425000$$

The profit is $\Pi = R - C$. 

The average cost is $C/12500$. 

I leave you to crunch the integrals.

Edit: In looking at your problem, $p(q) = 60 - 0.002q$ is definitely an inverse demand function. The revenue $R = qp(q) = (60q - 0.002q^{2})$ (quantity times prices). This yields the profit maximization problem (maximizing with respect to quantity produced):

$$\max_{q} qp(q) - c(q) = (60q - 0.004q^{2}) - 10q - 300000$$

The First Order Conditions are:

$$60 - 0.004q - 10 = 0 \implies 60 -0.004q = 10$$

You did the work from there to solve for $q$.