Stronger version(2) of Isolation Lemma for linear forms
For your first question, if $l=1$, then $\frac{K^\frac 23}\epsilon$ is sufficient. This follows the fact that the lower envelope of some lines has the same complexity as the upper convex hull of its dual point system and if you have a convex polygon with integer vertices in $[0,K]^2$, then it can have at most $O(K^\frac 23)$ sides. I know I did not explain the details, but this is anyhow only a partial answer to your question...