Random K-out graphs, denoted $\mathbb{H}(n;K)$, are generated by each of the $n$ nodes drawing $K$ out-edges towards $K$ distinct nodes selected uniformly at random, and then ignoring the orientation of the arcs. Recently, random K-out graphs have been used in applications as diverse as random (pairwise) key predistribution in ad-hoc networks, anonymous message routing in crypto-currency networks, and differentially-private federated averaging. In many applications, connectivity of the random K-out graph when some of its nodes are dishonest, have failed, or have been captured is of practical interest. We provide a comprehensive set of results on the connectivity and giant component size of $\mathbb{H}(n;K_n,γ_n)$, i.e., random K-out graph when $γ_n$ of its nodes, selected uniformly at random, are deleted. First, we derive conditions for $K_n$ and $n$ that ensure, with high probability (whp), the connectivity of the remaining graph when the number of deleted nodes is $γ_n=Ω(n)$ and $γ_n=o(n)$, respectively. Next, we derive conditions for $\mathbb{H}(n;K_n,γ_n)$ to have a giant component, i.e., a connected subgraph with $Ω(n)$ nodes, whp. This is also done for different scalings of $γ_n$ and upper bounds are provided for the number of nodes outside the giant component. Simulation results are presented to validate the usefulness of the results in the finite node regime.