Does the air resistance of a jet engine air intake count towards the jet's total air resistance?
This is a very tricky topic. Basically, you need a clear definition of what is part of the engine-related drag and what is counted towards airframe drag. Splitting up drag between components will always produce an arbitrary result.

An intake can generate thrust as well as drag. In case of the SR-71 at Mach 3, the intake contributed 54% of overall thrust, if you just look at the pressure distribution over the intake walls.

Generally, the job of the intake is to feed air to the engine compressor at just the right speed and as homogeneously as possible. This requires it to suck in air when at rest from all around the aircraft, and to ingest only the core of the stream of air flowing towards it at high speed. What is not ingested needs to flow around the intake lips and the engine fairing, and the drag caused by this process even has its own name, being called spill drag.

At high speed the intake needs to slow down the oncoming air and does so already ahead of the intake lip at subsonic speed. At supersonic speed this slowing down is accomplished by shocks, which adds another source of drag. This can be easily counted as part of the airframe drag, but done right this slowing down will compress the air, which in turn will increase thrust. Apart from viscous losses, this compression pays for itself and is almost drag-neutral in a good design and at ideal conditions.

Next is the flow around the intake. Just throttling the engine will change the airflow through it and thus the flow pattern around the intake. You can imagine that some configurations show marked changes in aerodynamic drag depending on the throttle setting. In case of the Eurofighter, the vicinity of the canard will even cause throttle changes to result in notable pitch moment changes.

To answer your question: Yes, the intake will produce drag, and how much it does depends on how you define drag. It is certainly wrong to think that the stream tube hitting the intake will just disappear into it and not add any drag.