Since $-5 < a, b, c, d < 5$ the inequalities $5< a+b< 10$ and $-10< c+d< -5$ show that
$$0< a, b< 5\quad\quad -5< c, d < 0$$

It is possible then to conclude that

$$10 < a+ b- c - d < 20$$
$$0 < a- c < 10$$
$$-10 < a - c + d - b < 10$$
$$0 < abcd < 625$$
$$0 < \frac{|a|+|c|}{2}-\sqrt{|ac|} < 2.5$$

Note that the lower bound of the fourth inequality could be deduced from the AM-GM inequality for two numbers.

Note also that, since it is not possible to set, for example, $a=5$ care must be taken to construct a really clear justification of the results.