Some improvements to the lower bounds appear in
Lemm, Marius, New counterexamples for sums-differences, Proc. Am. Math. Soc. 143, No. 9, 3863-3868 (2015). ZBL1358.42006.
In particular for 1, there are now examples with $|X| \geq N^{1.77898}$ and for 2 there are examples with $|X| \geq N^{1.61226}$. They build upon an earlier entropy-theoretic construction of Ruzsa that improved on the lower bounds in my original paper with Nets.
As far as I know there have been no improvement in the upper bounds (unless one also adds more slices to the hypothesis).
The recent paper
Green, Ben; Ruzsa, Imre Z., On the arithmetic Kakeya conjecture of Katz and Tao, Period. Math. Hung. 78, No. 2, 135-151 (2019). ZBL1438.42040.
forms some further equivalent forms of the various problems here, and gives some further applications, but does not numerically improve the exponents for the upper and lower bounds. I would imagine these two papers between them would pretty much summarise the current state of the art.