If it is really an emergency you should have already called 911. Maybe the dispatcher says drive as fast as you can cause we cannot get an ambulance to you - but in that case you stay on the phone. The dispatcher talking to you is also on radio with police and the flashing lights are your escort as the dispatcher will tell you.

But the above is something that I doubt will actually happen - ambulances are generally close.

Roads are engineered for a particular speed and flow of traffic. If you exceed the design limits of the road, you may feel very unsafe and be quite inclined to slow down again. This applies even to large, wide, noticeable emergency vehicles. Residential streets often participate in noticeable "traffic calming" measures, like roundabouts, speed humps, wasp waist narrowing, landscaping and trees, signage, etc.

Most of the time when I see emergency vehicles plowing down the street, they are pretty much going at the limit, but they look faster, because all the traffic signals favor them, and all the other motorists have pulled out of the way. But a fire truck has still gotta slow down or stop at a busy intersection or just a turn. Pedestrians are a thing. And scooters, and cyclists wearing headphones...

That depends on the emergency and the road. In Manhattan you cannot go as fast as the rural highways - but the only places you would go are also pretty close. In rural areas when seconds count the correct hospital may be hours away (rural hospitals cannot solve all problems), and those emergency personal will drive much faster than the road design (which is also designed much faster than Manhattan) if that is what it takes to get there.