Do most airports (especially busy ones) use only one runway at a time for landing? For example, I know Heathrow uses only one of its runways at a time (although they might switch to another half way through the day).
In general, any runway can be used for both takeoffs and landings. However, switching back and forth requires substantial safety margins. If you can use one runway for just takeoffs and another for just landings, efficiency goes up. If those runways are parallel, efficiency goes up again. This is why major airline hubs tend to have pairs of parallel runways.
Averaged over a day, the number of takeoffs and landings should be about the same, so the normal mode would be to split the runways as evenly as you can. If the number of runways isn't even, the last runway will often alternate throughout the day to mitigate short surges in one or the other.
As to which runways are used for what, the ones used for landing should be as far from each other as possible to maximize lateral separation of arriving traffic on instrument approaches. (Departing traffic mostly separates itself.) So, if you had four runways organized as two pairs, you'd expect the "inner" one of each pair to be used for takeoffs and the "outer" one of each pair for landings.
In the case of Heathrow specifically, runway usage is largely dictated by noise abatement agreements with nearby towns, rather than what makes the most sense operationally.
Airports will usually divide their active runways about evenly between takeoffs and landings. KATL (Atlanta Hartsfield Jackson) has 5 runways, and currently operates 2 for takeoff and 2 for landing. Most airports that can operate more than 2 runways simultaneously will use them all, unless the traffic is just very low.
Heathrow is exceptional in that it is one of the busiest airports in the world, and only has 2 runways.