DA20 pilot reported aircraft communication failure that resulted in two NMAC's while attempting to land. Pilot took evasive action to avoid collision for one NMAC.

Narrative

I recently was checked out in a DA20-C1 aircraft; an aircraft I have roughly 800 hours flying but the vast majority of that was over 15 years ago. On this day I was accompanied by my son for his first flight in a small plane. My first error was to not do a radio check and verify that the radio was transmitting. Reflecting back I realize that as a professional airline pilot this is not part of my normal flow; although it was when I flew GA. As a result I was unaware that when I made radio calls others couldn't hear me; although I could hear them. My son and I each had difficulty adjusting the sensitivity of our microphones but found a mostly satisfactory setting. Additionally; and more importantly; I found a high level of background noise while taxiing making it sometimes difficult to hear other transmissions; and noted that even turning up the volume to full some transmissions seemed low or unclear. I erred in deciding the noise level was acceptable; not thinking that the engine would provide more noise when at higher power settings than idle.We departed the pattern from runway XX after announcing our intentions. Another aircraft declared a go-around. I believe this aircraft had announced a turn from base to final and I believed we had more than sufficient room to depart before they reached the runway. Most critically; they didn't hear my radio call as we entered the runway. This was my first clue that something was wrong with my transmissions; although it only became apparent upon later reflection.After departing the pattern and flying south and east of ZZZ Airport I attempted to contact ZZZ1 Tower to enter their airspace; but after repeated attempts I received no response and elected to return to ZZZ airport. I assumed that ZZZ1 Tower was too busy to accommodate my request (there was a lot of traffic in their airspace at the time); but; again; this was another clue that others couldn't hear my transmissions. We then attempted to enter the traffic pattern on a 45 for ZZZ. After making a 5 mile call and a 3 mile call and while listening for other traffic we neared the downwind leg and realized another aircraft was too close for comfort--500+ feet away. Although making every effort to identify other traffic via their radio calls I evidently missed their call; likely because of the noise level in the cabin coupled with the low volume of the headset (I had it turned to max). We made a right 360 turn with the intent to re-enter the pattern. After two turns we entered the downwind leg for runway XX and began our descent for approach. Upon turning base we saw we were very close to another aircraft on final; perhaps 200 feet away; and I turned right; added full power and we exited the pattern. I was now feeling uncomfortably vulnerable; recognizing that I was either not hearing others' calls or was misjudging their position; thus my situational awareness was degraded.By this time my son informed me he was feeling increasingly airsick as we flew south and west to attempt another entry to the pattern. I continued to make radio calls as we began to enter a 45 for the pattern; recognizing it was unlikely anyone could hear us. As we entered the 45 another aircraft announced they too were entering the 45. Evidently they couldn't see us and we couldn't see them and we had no way to communicate our position. I elected to continue. Fortunately another aircraft recognized our plight and announced our position for us. The other aircraft near us chose to depart the pattern and we continued to base and final for a full stop landing.With the advantage of hindsight I recognize that the chain began with my oversight of making a radio check to test outbound transmission. I am still uncertain whether it was an equipment error or that I was unfamiliar with the radio (likely the second; although I troubleshot the radio once I realized my transmissions were unheard). My second error was deciding that I was able to hear others' transmissions sufficiently well even though it was less than optimal even when on the ground. I was careful to listen to others transmissions while flying and made every effort to determine others positions; especially while entering and flying in the pattern. However; given the noise level and the sometimes low volume coupled with my stress level I can only surmise that I missed understanding several calls; further compromising our safety and others'.My intent is to work with an instructor to be certain I am fully familiar with the radio; to make sure in all future flights to perform a radio check; and to always ensure that the noise level and volume level are optimal before departing.