Dick's Personality and Interactions with Others
Dick will still have the same personality as canon. The main difference is that he will also be much more secretive and mysterious because he protects his identity much more than canon. So while he does trust, there's a bit more of a barrier. (Not to mention that there's no Batman to compare him with).
Robin and later Nightwing, however, is still one of the most loved, respected, capable, and trusted heroes ever. Dick can also rely on some of his associates to help with crime in both Gotham and Bludhaven since he doesn't have the whole Batfamily.
Dick seems more flighty and unreliable as a civilian because his family and loved ones don't know that he is a vigilante/super hero. He seems like a carefree young adult, comes by randomly, comes late to some events if he can make it, sometimes misses events or leaves early, and doesn't have a steady job or living situation (to their knowledge). Dick never went to college because of his hero work and his problems with the Court of Owls. Dick attended the police academy, but dropped out instead of taking the exam. His reason for attending the academy was to get more training and gain insight into the corruption in the police force. Dick also met and made a connection with Amy, who he knows can be trusted and becomes his equivalent of what Jim Gordon to Batman. He works a lot of random jobs while undercover, like a barista, sales person, gymnastics tutor, coder, etc.
Despite his seeming unreliability, they do love Dick. They don't think he's mature or takes things seriously.
Once Bruce goes missing, Dick takes a bit of a sabbatical to protect Damian and help keep things together, only being on call for big emergencies and maybe some training. Dick also investigates where Bruce is, maybe following some leads from Tim. Alternatively, Tim could also be investigating where Bruce is on his own, deciding to leave home. Dick can keep an eye on him and keep him safe and alive, as well, without Tim ever knowing. His other siblings are surprised and maybe kind of envious that Dick stays around to take care of Damian. The others can also still be doing their own thing in their grief, not really being around.
After Bruce is rescued, returned and recovered, Dick gets back into his hero duties. He does make sure to keep up with Damian. Damian can also stay with Dick on weekends. They can have a signal for emergencies.
Dick can also have trouble with romance because of his double life. He will have very few relationships because of his workload. Dick will have feelings for/fall in love with Barbara Gordon, Starfire/Kori'andr, and maybe Bea Bennett. Unfortunately, these relationships will not work out because of his secret identity along with other factors. Because of this, he'll mostly focus on saving the world, protecting his loved ones from harm, and dealing with the Court of Owls.