Thanks for the question! In most situations, sedimentary rocks would metamorphose as they are heated and the metamorphic rock would begin to melt when the temperatures are high enough. There are only three ways that I know of in which a sedimentary rock could be heated up fast enough to melt without ever becoming a metamorphic rock... and all three of them are amazing:

-Meteorites hit the Earth so fast that they can instantaneously melt rocks that they collide with, the small pieces of igneous rock produced this way are called tectites. Of course, the larger the impact, the higher the energy and the more magma can be made. Perhaps the best example of this happening at a large scale is in the Sudbury basin in Canada.

-Earthquakes can actually cause rocks to melt. Earthquakes happen with the force of the moving teconic plates becomes greater than the strength of the rocks, causing the rocks to fracture and slide past each other for some short distance (10's of feet for the absolute largest earthquakes, but usually much less). As the rocks are sliding past each other along the fault, the friction is so strong that it can actually melt the rock within a few centimeters of the fault, this melted zone (technically an igneous rock) is called a pseudotachylite.

-Lightning strikes actually melt the ground they strike, by flash heating it to over 3000 F. The type of rock formed this way (never more than a few inches large) is called a fulgurite (these can be spectacularly shaped, I recommend doing an image search for some).

Tectites, pseudotachylites, and fulgurites are all technically igneous rocks, however they are definitely odd-ball rock types. Usually sediments have to metamorphose before they can melt.

Let's start with the definition of an igneous rock -- a rock forming from magma. Metamorphic rocks are any kind of rock whose minerals recrystallize due to either high pressures, temperatures or both. Often when rocks are highly metamorphosed there is some degree of melting happening but the melt has a composition of melted sedimentary rocks, not of rocks that came from magma.

There are also igneous rocks that are metamorphosed. They may also slightly re-melt and this melt will be of igneous rocks but the textures will be different. When the minerals re-crystallize they will no longer resemble an igneous rock (they crystallize differently), so we call them metamorphic.

For rocks to be igneous they must be fully recycled -- either exploding out of a volcano (extrusive) or forming from an underground magma system (intrusive).

Yes: metamorphism is a consequence of pressure as well as heat. It is easily possible for a rock to be heated to the point where it melts without being subject to the pressure needed to cause it to metamorphose.

For example, suppose a volcano erupts basalt lava. Basalt has a higher melting point than granite. Now suppose this molten basalt touches a piece of granite that eroded out of a mountain. The granite will start to melt. This is despite the fact that the granite is at only one atmosphere of pressure.