Reconfigurable Intelligent Surfaces (RISs) can redirect electromagnetic waves to desired directions to enhance signal coverage and/or improve signal-to-noise ratio (SNR) at the user equipment (UE). We present the design, implementation, and evaluation of an RIS-assisted O-RAN 5G system operating in the FR2 millimeter wave (mmWave) frequency band. We first introduce the design of 1,024 element (32 $\times$ 32) 1-bit RIS operating at the 28 GHz band, utilizing a modular and scalable tiled architecture. Then we demonstrate how the O-RAN E2 interface can be leveraged to dynamically control RIS configurations without modifying standard 5G signaling procedures. To evaluate the RIS-assisted 5G system, we conducted extensive field trials in both indoor and outdoor environments. The results of the O-RAN link coverage trials show that the deployed RIS provides substantial received signal power gains, ranging from 9 to 20 dB and 6 to 18 dB in indoor and outdoors scenarios, respectively. Handling UE mobility in RIS-assisted systems is challenging due to the need for joint RIS and UE beam management. For that, we develop two UE mobility management algorithms and evaluate them in real-time operation using the RIS O-RAN testbed. These algorithms leverage the received signal power at the UE to jointly track and adapt the RIS and UE beams in real time as the UE moves. The findings draw important insights into the practical feasibility of integrating RIS into O-RAN systems to enhance coverage, mobility support, and link reliability in next-generation cellular networks.