Expected value of sum of two dependent Binomial variables

Let's consider an equivalent experiment. We have $n$ identical coins, numbered from $1$ to $n$. We toss each of the $n$ coins once; then each coin which came up tails is tossed one more time. Let $X_i$ be the number of times the $i^\text{th}$ coin comes up tails, and $X=\sum_{i=1}^nX_i$ the total number of tails obtained. $$E(X)=\sum_{i=1}^nE(X_i)=\sum_{i=1}^n(p+p^2)=n(p+p^2)$$

To calculate $E(T_2)$, consider using the law of total expectation.

$$E(X) = E\left[E(X|Y) \right]$$

Which in your case would give

\begin{align*} E[T_2] &= E\left[E(T_2|T_1) \right] \\ &=E\left[p \cdot T_1 \right] = p E\left[T_1 \right] \\ &= np^2 \end{align*}