1. When the rear of the camera is opened, any film that gets exposed to light is ruined. Any film that is concealed from light – and I mean really concealed – is saved.

2. When you use the remote, listen for the sound of the shutter and film advance mechanism. Check the frame counter too – if it goes up by one, then a frame has been exposed. If it hasn't gone up, then check/change the battery in the remote.

3. Most automatic cameras will reset the frame counter when the rear of the camera is opened. When you close the rear again, the camera usually assumes you've put a fresh roll of film inside, and it should advance the film by a small amount to ensure that unexposed film is sitting behind the lens.

I would say... continue using this roll until the end is reached, but maybe best not to use it for anything critical. Get it developed (or do it yourself). Take note of how everything turns out, and just use it as a learning experience if nothing else. If a lab is developing it, it's probably worth mentioning to them in advance what you've done.