Internal pressure is almost certainly a safety issue and battery failure mode. Same for anything leaking out. Even if the substance entered the phone and not something decomposing from the materials made into the phone, you don't want electrical shorts or anything leaving the case of an iPhone.
Think potential fire hazard if the battery is swelling or leaking and get the hardware to someone that is qualified to diagnose a failing battery and physical damage. Call Apple if you can't take it to someone for service. 1-800-APL-CARE
Do not charge it or connect to any USB - I would power it off immediately from the description you have listed.
You can also get safety information from: http://www.apple.com/us/search/iphone-safety?find=iphone+safety and select Support to narrow the results to specific hardware safety and manual download instructions.
Once you have some guidance, you might power it on one more time to perform a backup if you don't have a recent one. Again, don't leave it unattended until you're sure it's stable and working correctly. If you can't get it cooled, consider calling for help - local fire department if needed.
You could use Audio Hijack (per Tetsujin) to capture the audio as needed or you could use a USB audio adapter to do it for significantly less.
I use the Behringer UCA202 Audio Interface with my iMac.
When I needed to record a screencast with the audio from the application, I just used an RCA audio cable I had laying in a drawer similar to the image below. You don't need anything fancy or expensive, just a standard RCA audio cable.
In quicktime, I just selected the USB device as my input:
Now, when I say significantly less, Audio Hijack is about $50 if buying the new license. The USB Audio device is $30 and if you have to buy an RCA audio cable, you're looking at another $5. So, for $35, you get an audio interface that has many more uses than the software can provide (for example, I can hook my Bose Wave Radio to my iMac now) whereas the software doesn't allow you to do that.
Make no mistake, Audio Hijack is good software. This is just another, albeit less expensive, avenue to solve your problem.
Best Answer
The screenshot will include whatever is supposed to be there.
For example if you're on your home screen it will show the part of your wallpaper you cannot physically see when looking at the device.
If you're watching a video horizontally on YouTube for example, it will show whatever is supposed to be displayed in the notch too.
While watching videos horizontally you actually lose out on what is supposed to be displayed in the notch, rather than scaling the video size down to accommodate what would be a smaller res video. If something vital was displayed in the non-visible notch part of the video, you could take a screenshot of the frame, view the part you needed, then delete the screenshot without saving it for example.
You can see an example of a screenshot on an iPhone X here.