I've recently been made responsible for watering and taking care of many small potted plants around our office, and I have little to no gardening experience. There are many different types of plants, and I don't have the time to research the details of each one. Is there a simple rule of thumb for how much/often to water plants like these, in general (and how to tell whether they're being over or under watered)? What about other maintenance (e.g. cutting or removing dead leaves)?
If a house plant is in reasonable light then when it is watered it should receive enough so that water comes out the bottom of the pot. If a plant has really dried out then you may need to let it sit in a small amount of water so capillary action can re wet the entire root ball. Normally though you should not let a plant sit in water.
Although it varies from species to species as a general rule of thumb if the leaves are turning yellow and falling off while plump with water then you are over watering. Under watering can produce yellow leaves but they tend to be dry and papery.
The primary cause of death of houseplants is over watering, with inadequate light a close second. (I base this on my 10 years as an interior landscaper but articles here and here seem to confirm this.)
Smaller pots dry out faster and the more sun the more water a plant can use. It is difficult to give a one size fits all rule. Put your finger in the soil or use a moisture meter, if it's dry water adequately. If it's wet, leave it be.
Yellow leaves should be removed at the base of the stem by hand. Be careful not to damage the main stem. I have found that by cutting only the leaf leaves a bit of the stem that has to be removed later anyway. So if you are going to trim a leaf, it all has to go.