Can I safely run the microwave for an extended time with just a bowl of water?
A bowl of water is the standard load for testing microwave ovens. Use a microwave-safe material for the bowl. Simple plastics like polyethylene or polypropylene are OK, not melamine which will overheat. Glass is OK, as are most plain ceramics, but some decorative ceramic glazes can overheat.

It's not so much that the absorption of microwave energy by the water stops the magnetron overheating (it's the noisy fan that does that), but that it prevents reflected energy causing high voltages at the magnetron output which could destroy it.

Use a large enough bowl so the water doesn't boil, some microwave ovens (like mine) do a rubbish job of ventilating steam from the cavity, leading to a lot of condensation. Condensation in the door seal may well alter your microwave leakage measurements.

As a rule of thumb, it takes more than 5 times as long to boil a quantity of water away, as it does to heat that quantity from room temperature to boiling. That will give you an idea of the safety margin for heating time. If the bowl boils dry, then you've lost the protection of the water.

As MrPhooky and Robherc have pointed out, it's possible to superheat water in a microwave under sufficiently clean conditions. If there are no nucleii to initiate bubbles of steam, the water could exceed 100C. If you then disturb the bowl, the water could flash boil, dumping all of its excess heat out as steam, very dangerous. This happens rarely, but once in your face is once too often. Mitigations are to a) make sure the heating time and water quantity take you nowhere near to boiling or b) use boiling stones or some other source of boiling nucleii in the water (a bit of eggshell, or a carrot top). Tap water is usually impure enough to be OK, but don't bank on it. Robherc linked this video which shows flash boiling.