In this paper, we address the analysis and the output-feedback synthesis problems for linear parameter-varying (LPV) sampled-data control systems with potentially variable sampling rates. We assume that the state-space matrices of the plant and the sampling interval depend on parameters that are measurable in real-time and vary in a compact set with bounded variation rates. We explore criteria such as the stability, the energy-to-energy gain (induced L-2 norm) and the energy-to-peak gain (induced L-2-to-L-infinity norm) of such sampled-data LPV systems using parameter-dependent Lyapunov functions. Based on these analysis results, the corresponding sampled-data output-feedback control synthesis problems are examined. Both analysis and synthesis conditions are formulated in terms of linear matrix inequalities (LMIs) that can be solved via efficient interior-point algorithms.