Swedish citizen going from Copenhagen(Denmark, closest major airport) to Vietnam via Beijing.

1. You entire transit is covered by 24 hour visa free transit, so it is a status and not a visa. In order to leave the airport you will apply for a temporary entry permit immediately upon your landing in Beijing. You will fill out arrival/departure card on board the plane that will take you to Beijing. When you exit the plane the first thing you will do is to approach 24/144 hour visa free transit counter where you will apply for a permit by presenting your flight tickets. Singe your total transit time is relatively long, your chances to be granted a permit are very good. After your permit is granted, you will proceed toward immigration in order to enjoy in some sightseeing.
2. This is absolutely free of any charge.
3. If you have bought your journey under one booking, your luggage will be tagged all the way to Vietnam and you do not have to bother yourself about it. If you travel on two separate bookings/tickets, they you will collect your luggage, store it at left luggage facility that exists at the airport and proceed for a sightseeing. I am more than certain that you have booked your journey as one ticket/booking, so you will not have to worry about this detail at all.