How to work with distributions that are discrete in some variables and continuous in others? 
So, in this problem I am asked to find the cumulative distribution of the continuous variable T and the probability that $J=3$ given T is larger than 10 and lastly the conditional expectation of T given $j=2.$ I have tried just adding the pdf but they do not have the same range? Any idea what to do?

As many comments pointed out, the information given by "$f(t, j)$" needs more clarification, because it is neither a simple pdf/pmf nor a standard representation of mixture model.  My interpretation to $f(t, j)$ is that it characterizes the joint distribution of the random vector $(T, J)$ in a way such that (in particular, you should NOT interpret $f(t, j)$ as the conditional density of $T$ given $J = j$, as $\int_\mathbb{R} f(t, j)dt < 1$ for all $j$).
\begin{align}
& F_1(t) := P[T \leq t, J = 1] = 
\begin{cases}
0 & t < 0, \\
\int_0^t f(x, 1)dx = \frac{1}{300}t^2 & 0 \leq t < 5, \\
\frac{1}{12} & t \geq 5.
\end{cases} \\
& F_2(t) := P[T \leq t, J = 2] = 
\begin{cases}
0 & t < 0, \\
\int_0^t f(x, 2)dx = \frac{1}{600}t^2 & 0 \leq t < 10, \\
\frac{1}{6} & t \geq 10.
\end{cases} \\
& F_3(t) := P[T \leq t, J = 3] = 
\begin{cases}
0 & t < 0, \\
\int_0^t f(x, 3)dx = \frac{1}{432}t^2 & 0 \leq t < 18, \\
\frac{3}{4} & t \geq 18.
\end{cases} \\
\end{align}
It can be verified that for $t \geq 18, F_1(t) + F_2(t) + F_3(t) = P[T \leq t] = 1$, therefore, this interpretation, i.e.,
\begin{align}
f(t, j) = \frac{dP[T \leq t, J = j]}{dt}
\end{align}
indeed allows for a valid marginal distribution of $T$.  In the literature, $F_j(t)$ are sometimes referred to as "sub-distribution function".  The marginal distribution function of $T$ now can be obtained by adding $F_j(t)$:
\begin{align}
F(t) := P[T \leq t] = 
\begin{cases}
0 & t < 0, \\
\frac{79}{10800}t^2 & 0 \leq t < 5, \\
\frac{1}{12} + \frac{43}{10800}t^2 & 5 \leq t < 10, \\
\frac{1}{4} + \frac{1}{432}t^2 & 10 \leq t < 18, \\
1  & t \geq 18.
\end{cases}
\end{align}
I will leave the remaining two calculations ($P[J = 3 | T > 10]$ and $E[T | J = 2]$) back to yourself, which are not hard to derive based on the above discussion.  You shall also try to recover the standard representation of $F(t)$ in terms of a mixture model, i.e., try to write $F(t)$ as
\begin{align}
F(t) = p_1 P[T \leq t | J = 1] + p_2 P[T \leq t | J = 2] + p_3 P[T \leq t | J = 3],
\end{align}
where $p_j = P[J = j], j = 1, 2, 3$.
