To produce a pressure from the trumpet, we need to apply a pressure into the trumpet/tube. For most of us, this means we must blow air into the tube. But, the tube must resist the air we apply: else no pressure will result.
Not exactly. Our embouchure must resist air movement if vibration is to result, but the pressure waves from that vibration perpetuate in air thanks to the inertia of that mass of air, not any raceway resistance to flow. Raceway backpressure just makes it easier for us to produce vibration with our lips.
To produce a frequency, we need to apply a vibration to the tube: for us, the source is our lips. For most of us, we need to exhale air against closed lips to produce a vibration/frequency.
Yes, although trumpet tone is the application of that vibration to the air column shaped by the tube, not so much the brass tube itself.
Unfortunately, the science we learn in high school is erroneous, and thus people carry this throughout their lives. Sound is always depicted via 2-dimensional graph as a line to form a 'wave'; this pictorially presents sound to have a frequency we can measure, but excludes entirely the factor of pressure. But here is a simple fact: sound is 3-dimensional and 2-factored.
Perhaps over-simplified would be a better term. 2-dimensional representation of pressure over time is not in itself erroneous.
What we hear as high pitch from a trumpet is a function of both frequency and pressure: ie greater vibration and greater pressure, from the player. To achieve this, many factors must be combined. One factor is the resistance of the trumpet to help us create greater pressure. Thus, a trumpet requires bends in the pipes; the bends assist to create pressure. If the right pressures are achieved, what we call 'partials' can result. This is why a trumpet having a D-shaped lead pipe will be easier to play for high notes than one having a semi-circular C lead pipe.
Pitch is a function of the periodic frequency of pressure variation. Greater frequency creates higher pitch. Greater pressure creates higher volume (the auditory meaning of volume, not the geometric). The properties of square, D and single-radius slides impact the way in which the energy damping functions of both the structure of the instrument, and the transition from laminar to turbulent flow and back again, require us to add more energy, more intense vibration, to the air column at a given pitch, or as we change pitch. These can particularly affect the ease with which by altering the input spectrum, that pink noise of frequencies produced at the embouchure, we shift the natural resonance of the system to bend notes. The less abrupt the geometry, the easier bending is to achieve generally (but never count on anything as complex as a trumpet behaving consistently in response to a single variable given the intricacies of design)
For anyone who may doubt the above, kindly ponder why the flame produces a sound exiting the glass tube; when the flame itself consumes air and burns without a vibration.
The air is not consumed. Oxygen is consumed and replaced primarily with carbon dioxide. The mass and volume of the air actually increase as a result of the addition of the carbon (and heat).