This paper describes continuous-space methodologies to estimate the collision probability, Euclidean distance and gradient between an ellipsoidal robot model and an environment surface modeled as a set of Gaussian distributions. Continuous-space collision probability estimation is critical for uncertainty-aware motion planning. Most collision detection and avoidance approaches assume the robot is modeled as a sphere, but ellipsoidal representations provide tighter approximations and enable navigation in cluttered and narrow spaces. State-of-the-art methods derive the Euclidean distance and gradient by processing raw point clouds, which is computationally expensive for large workspaces. Recent advances in Gaussian surface modeling (e.g. mixture models, splatting) enable compressed and high-fidelity surface representations. Few methods exist to estimate continuous-space occupancy from such models. They require Gaussians to model free space and are unable to estimate the collision probability, Euclidean distance and gradient for an ellipsoidal robot. The proposed methods bridge this gap by extending prior work in ellipsoid-to-ellipsoid Euclidean distance and collision probability estimation to Gaussian surface models. A geometric blending approach is also proposed to improve collision probability estimation. The approaches are evaluated with numerical 2D and 3D experiments using real-world point cloud data. Methods for efficient calculation of these quantities are demonstrated to execute within a few microseconds per ellipsoid pair using a single-thread on low-power CPUs of modern embedded computers