When the washer pump engages, the wipers shouldn't stop.
I'd want more information as to when the fuse blows. Do the wipers have anything to do with blowing the fuse?
As I said, I'm not too familiar with the Honda electrical system. In my Toyotas, the power windows have a circuit breaker, which can be reset, though it's a pain to get to the breaker. (Same breaker also covers the power seats.)
Fuses only pop for a few reasons. The most common is someone poking around and accidentally grounding something while working on the car, but that's not a recurring thing.
The next would be a shorted wire. These are usually where there's a pinch point, or a through-hole where a wire passes through a panel between supply and accessory. These can be harder to locate because the wire by necessity passes close to the panel, usually inside a grommet which can appear to be okay on a cursory inspection. These will fall into a category of a fuse that will last for a while, then blow, or will blow in conjunction with a physical shock, like the car going over a bump, or opening/closing a door, or moving a panel.
Sometimes, other things have been tapped into circuits. Aftermarket accessories like remote starters, alarm systems, or audio systems can be tapped into an inappropriate circuit and increase the draw on a circuit beyond what the fuse capacity is. Often times, the fuse blows when something it turned on (or turned up.)
The last situation where fuses go is where a component is broken down and taking more power than it should. In the case of the instrument panel, this is unlikely unless you're also seeing something unusual on the dash. The indicator lights are either on or off. The gauges use a variable amount of current, but there should be enough capacity for their normal range of current draw.