In this paper, we explore the integration of communication and synthetic aperture radar (SAR)-based remote sensing in low Earth orbit (LEO) satellite systems to provide real-time SAR imaging and information transmission. Considering the high-mobility characteristics of satellite channels and limited processing capabilities of satellite payloads, we propose an integrated communication and remote sensing architecture based on an orthogonal delay-Doppler division multiplexing (ODDM) signal waveform. Both communication and SAR imaging functionalities are achieved with an integrated transceiver onboard the LEO satellite, utilizing the same waveform and radio frequency (RF) front-end. Based on such an architecture, we propose a transmission protocol compatible with the 5G NR standard using downlink pilots for joint channel estimation and SAR imaging. Furthermore, we design a unified signal processing framework for the integrated satellite receiver to simultaneously achieve high-performance channel sensing, low-complexity channel equalization and interference-free SAR imaging. Finally, the performance of the proposed integrated system is demonstrated through comprehensive analysis and extensive simulations in the sub-6 GHz band. Moreover, a software-defined radio (SDR) prototype is presented to validate its effectiveness for real-time SAR imaging and information transmission in satellite direct-connect user equipment (UE) scenarios within the millimeter-wave (mmWave) band.