Object detection in biomedical settings is fundamentally constrained by the scarcity of labeled data and the frequent emergence of novel or rare categories. We present FSP-DETR, a unified detection framework that enables robust few-shot detection, open-set recognition, and generalization to unseen biomedical tasks within a single model. Built upon a class-agnostic DETR backbone, our approach constructs class prototypes from original support images and learns an embedding space using augmented views and a lightweight transformer decoder. Training jointly optimizes a prototype matching loss, an alignment-based separation loss, and a KL divergence regularization to improve discriminative feature learning and calibration under scarce supervision. Unlike prior work that tackles these tasks in isolation, FSP-DETR enables inference-time flexibility to support unseen class recognition, background rejection, and cross-task adaptation without retraining. We also introduce a new ova species detection benchmark with 20 parasite classes and establish standardized evaluation protocols. Extensive experiments across ova, blood cell, and malaria detection tasks demonstrate that FSP-DETR significantly outperforms prior few-shot and prototype-based detectors, especially in low-shot and open-set scenarios.