CE-700 First Officer reported receiving a TCAS RA and an EGPWS alert while maneuvering for being high on the approach.

Narrative

We were cleared from I believe 15;000 to 5;300; then cleared visual Runway XX. We were VPATH captured with the visual [Runway] XX loaded in the box from straight. I assumed that would give us vertical guidance to final. As we were getting closer ATC called out the airport distance and direction. Based on what our VPATH was indicating I was looking too far out. ATC offered us vectors for descent - I assume because they knew we were high. We declined thinking we were set up well; planning on giving ourselves a direct to straight and flying the whole thing in automation.We finally saw the airport and realized we were high. We got cleared for the visual and ATC cleared us to maneuver in the airspace. I did not want to turn inside of the straight-in; so I decided to do a 270 left turn away from the airport to intercept the final approach outside of the straight-in for a more manageable descent angle / rate. I'm still in automation in heading mode at this point. The box had already sequenced to final. On our turn outbound; we get a TCAS RA at the same time ATC tells us there is a King Air maneuvering in that vicinity. I see the icon on the MFD (Multi-function Flight Display) and immediately make visual contact and kick off the autopilot. Given the fact that we were already Flaps 2 in a 25-degree bank and I could see we were turning in opposite directions and there was no danger of colliding with the King Air; I continued my descending turn following the command bars. However; since the box had already sequenced to final; it was telling us to go too low for that distance. We received an EGPWS for descent rate and then terrain. The terrain is relatively flat and neither one of us saw anything of concern. I leveled off and the warnings stopped so we continued the approach uneventfully. The whole event was 5 - 15 seconds long in my estimation.Suggestions - I should've put an altitude at the airport for descent planning. I should've accepted the vectors. I should've leveled off sooner.