Is it true that a tarp shouldn't be covering the entrance of a snow-cave while sleeping in one? What's the reasoning to that? Is using a sleeping bag in them just fine?
You want to leave enough room at the entrance for airflow. If the entrance is totally blocked you could possibly suffocate on the CO2 you breathe out.
In addition to the entrance, keep an eye on the air exit at the top so it doesn't get blocked.
You can heat the inside significantly with just a candle.
To answer the sleeping bag question: Snow shelters drip. Constantly. You can deal with some of the drips by placing your ungloved finger on the drip and then moving down to the bottom of the wall. The heat of your finger melts a tiny channel and encourages the drip to follow it. Still, you will get damp, so the best approach is either to have a water-repellent sleeping bag (lots of alpine ones have this) or to put your sleeping bag inside a bivvy bag.
Build your sleeping platform high, ideally higher than the entrance. Build a "cold sump" so the cold air flowing down has somewhere to go.
The following diagram (found at the top of a Google search) shows snow caves but the concept is the same for a shovelup or an igloo.