Duopoly vs Collusion (quadratic costs) Suppose there are 2 firms; Demand curve is given by $P=1400 - 5(q_1+q_2)$ and cost function is given by $C_i = 5q_i^2$.

For cournot, the best response function comes out to be $q_i=70 - 0.25q_j$ giving $q_1=q_2=56$ and $P=840$. Thus $\pi_1=\pi_2$= $840 * 56 - 5*(56)^2 =31360$. 

For collusion $MC=MC_1+MC_2=10q_1+10q_2=10Q$. Thus $MR=MC$ gives $1400-10Q=10Q$ which in turn gives $Q=70$ and $q_1=q_2=35$. And $P= 1400 - 5(70) = 1050$. Thus $\pi_1=\pi_2=1050.35 - 5(35)^2 = 36750-6125=30625$.

What I don't understand is, how is profit greater in Cournot duopoly as compared to a collusion. Is it possible? Or did I make a mistake somewhere during the calculation of $MC$ for collusion?
Your computations are OK, but this is a trick question

The total quantity set by the colluding firms and the quantity chosen by  a monopolist are not the same in this question. By setting $MR = MC$ you are solving the profit maximization problem a monopolist instead of the profit maximization problem of the colluding firms. 

The reason for the difference is that the marginal cost is increasing over the entire domain of the production function. Suppose the monopolist wanted to produce $Q=10$. The cost would be $5 \times 10^2 = 500$. If each of the colluding firms produced 5 the total cost of producing 10 units would be $5 \times 5^2 + 5 \times 5^2 =250$. I.e. the two colluding firms have smaller diseconomies of scale than the monopolist.

The most efficient way to produce any quantity for the cartel is to split production between the colluding firms equally. In this case the marginal costs of the cartel members are equal. Shifting a unit of production from firm 1 to firm 2 will make producing that unit more expensive, as the marginal cost increases in $Q$.

Viewed through the  $MR = MC$ argument that you used, the cartel of two colluding firms has access to a lower cost function unavailable to the monopolist: $C(Q) = 5\big[\big(\frac{Q}{2}\big)^2 + \big(\frac{Q}{2}\big)^2\big] = 5\frac{Q^2}{2}$. 

Thus the cartel's profit function  is $\pi (Q) = (1400 - 5Q)Q - 5\frac{Q^2}{2}$. $MR(Q) = 1400 - 5Q - 5Q = 1400 - 10Q$, $MC(Q) = 5Q$
and $MR(Q) \overset{!}{=} MC(Q) \Rightarrow 1400 = 15Q \Leftrightarrow Q^* = 93\frac{1}{3}$. The total profit of the cartel is $\pi(Q) = 65333\frac{1}{3}$. The profit of the individual cartel member is $\frac{\pi(Q)}{2} = 32666\frac{2}{3},$ which is larger than the Cournot equilibrium you correctly obtained.

The monopolist, as described in your solution has a profit function of 
$\pi(Q) = (1400 - 5Q)Q - 5Q^2$ For him $MR(Q) = 1400 - 5Q - 5Q = 1400 - 10Q$, $MC(Q) = 10Q$. And $MR(Q) \overset{!}{=} MC(Q) \Rightarrow 1400 = 20Q \Leftrightarrow Q^* = 70$. I.e. the monopolist stops producing earlier, as its higher $MC$ prevent it from making profits at larger quantities.