A C210 pilot reported an unstable approach and bounced landing so they executed a go around. During the climb out they were involved in a NMAC with traffic in the vicinity.

Narrative

I filed to depart ZZZ1 direct to ZZZ in a Cessna T210L. My flight was received and accepted. ZZZ1 and ZZZ were both reporting visual metrological conditions with unlimited visibility and clear skies. I departed ZZZ1. I cleared the airport area and contacted ZZZ1 Departure. 3 [Departure] cleared me to ZZZ as filed with an initial heading and altitude assignment after which I was cleared direct to ZZZ. ZZZ1 Departure then handed me off to ZZZ Approach. After making contact with ZZZ Approach; I requested the RNAV Rwy XX at ZZZ. The Air Traffic Control received my request and gave me altitude and heading assignments. As I approached ZZZZZ; the Initial Approach Point for the approach; the Controller stated that I was to maintain 3000 feet and fly the inbound approach course. Though in my experience this was a non-standard instruction and practice; I read back the instruction and complied due to increasing radio congestion. Once on the approach course and still maintaining 3000 feet; I asked for confirmation that I was cleared for the approach. The Controller's response was stepped on and I only heard him state; 'cancel your IFR flight plan.' This confused me as a response to my question confirming whether I was cleared for the approach. I asked him to 'say again.' He clarified that I was 'cleared for the approach and I am to cancel my flight plan in the air or on the ground.' I chose to cancel with him in the air at that time. The delay to ensure I was cleared for the approach placed me above glide slope and distracted me from deploying my first notch of flaps when I extended my landing gear. These two things in combination caused me to be too high and too fast creating an unstable approach. In order to correct this; I used an excessive length of the runway to slow and stabilize the aircraft prior to the flare. Recognizing the amount of runway passing; I inadvertently bounced the aircraft and immediately decided to go-around. I added power; initiated a climb and then began a left turn to avoid entering the Class C airspace. During the climbing turn; I saw my airspeed drop rapidly to approximately 65 knots. I immediately went wings level and reduced my pitch then proceed to enter left downwind for Runway XX. The radio congestion was continual; so I decided to depart the airport and go to the next destination; ZZZ2. While in downwind; I cleared the navigator of the previous flight plan; however; I could not clear the 'direct to Runway XX.' Instead of continuing with the navigator; I focused on hand flying the aircraft out of the airport vicinity. At which point; I could see multiple traffic targets present in my immediate vicinity. In seeing and avoiding these aircraft; I took evasive action and continued my climb. In one case; I passed approximately 300 feet over one aircraft. In prioritizing see-and-avoid; I neglected communicating with ZZZ Departure and I inadvertently entered into the Class C airspace approximately 2 nm from the northern rim and reached approximately 4000 feet before exiting. Contributing factors; I believe; included: (1) a non-standard approach practice; (2) radio congestion; (3) an unstable approach; (4) an unusual attitude; (5) a navigator programming error and (6) compressed airspace for VFR traffic. I do not believe qualifications or competency of myself or the Air Traffic Controllers were contributing factors. Corrective actions; I believe; are to not cancel my flight plan in the air; rather to do so when on the ground. I will then be more apt to communicate with Air Traffic Control in the event of a go around.Perceptions and judgements on my behalf included the altitude shelf of the Class C airspace and an inability to communicate with ZZZ Departure. My perceptions of each were exacerbated by fear of a mid-air collision while operating in see-and-avoid conditions. Decisions on my behalf were to ensure the safety of the flight; therefore; I overlooked the altitude shelfof the Class C airspace; and I prioritized aviating and navigating prior to communicating with Air Traffic Control. Factors affecting the quality of my performance was a combination of events that individually would not have caused concern. The combination of events following the unstable approach included the bounced landing; unusual attitude; navigator programming error; and see-and-avoid reaction. Actions or inactions on my part and; I believe; the Air Traffic Controller's part were inadvertent and not deliberate. Further; actions or inaction on my part and; I believe; the Air Traffic Controller's part were not due to a lack of qualifications or competency.