Any barrel will get hot, it depends on how thick the barrel and how fast you're spitting the rounds out. Rule of thumb is that thicker barrels heat up slower but most sporting rifles have thin(ner) barrels which heat up fairly quick so 3-5 rounds followed by cool-off period is recommended. WRT eye relief, it varies from manufacturer/model but is generally 3 to 4". Different mounts/bases are available so that with few exceptions, scopes can be mounted to accomodate the eye relief. You'll want to mount as close to the bore as possible so consider the height of the rings as well.
Good luck and safe shooting.