The YouTube app (I'm assuming you're using the app, and not the mobile site) doesn't have its own volume setting. It uses the main Media volume control of the phone. When you're playing a video, the phone's Volume buttons should control the video's volume.
What do you mean by videos that are sent to you? Do you mean that someone is sending you a YouTube link via email or SMS, and you're tapping that link which then opens up the YouTube app?
I realize that your phone is your only device, but there must be some way you can access another device, like at your local electronics store, or perhaps at your local public library?