Group images are exact kernel quotients and preserve affine smooth connected properties

By [F1] form $P=G\mathrm{/}K$ and factor $f$ through $\bar{f}:P\to H$. Its kernel is trivial: a point of that kernel lifts fppf locally to a point of $G$; the lift lies in $K$, so its quotient point is the identity, and this equality descends. Thus [F1] makes $\bar{f}$ a closed immersion. The map $G\to P$ is faithfully flat, hence schematically dominant; consequently the scheme-theoretic image of $f$ is exactly the closed embedded $P$. This identifies $I\cong P$ and gives the asserted exact projection. Applying [F2] gives each of the three inherited properties.

For the normality assertion, every scheme-valued point of $Q$ lifts fppf locally to $G$, and every point of the image of $N$ lifts fppf locally to $N$, by step 1.1 applied to $N\to Q$. On a common refinement, their conjugate is the image of $gn{g}^{-1}$, which lies in $N$ by normality. Membership in the closed image subgroup descends on covers by vanishing of its defining ideal. Hence conjugation in $Q$ preserves that image as a subgroup scheme. AC is inherited from [F1]–[F2].