For capturing single events, turn the sweep to "single" instead of "normal" or "auto". Single will start recording when a trigger happens, and then it will stop, and disable further triggering. You'll have to push the "run/stop" button if you want to try to capture another try. But the nice thing about single sweep is, your captured sample is safely frozen unless you DO push run/stop again.
Be sure to turn on long memory if you want to capture the longest time period with the most detail.
You can adjust sample rate to trade off the recording time for detail. I've only got an old DS1052E, so I'm not sure if the details will be the same for you, but on mine, if you go into the horizontal menu, you can see the sample rate as you turn the horizontal scale knob. You want to capture at least a few samples for each bit. Note that the scope captures samples with much more horizontal resolution than it displays on the screen during recording. You can set the scale to get the desired sample rate, then capture the sample, and you may not be able to see much detail. But after the sample is captured, you can use the horizontal scale and position knobs to zoom in and pan around and see more detail.
You can save a .wfm file to the scope's internal storage, or to a USB drive. There are various programs to view .wfm files, though the scope itself is often good enough.
Regarding the input and output showing the same thing, that's probably a side effect of the way many radios communicate with a computer for uploading/downloading frequency lists. They use a single line for both sending and receiving. It's got weak pull-ups on each end, and each end has a strong pull-down, along with a buffer that's connected to a UART. If you're looking at the RS-232 side of things, you've got a separate lines for TXD and RXD. Everything set on TXD will show up on RXD, but things received from the other end will only show up on RXD. Those interested in more details can Google for "Icom CI-V interface" (a similar idea is used by many radios, not just Icom).