Your question: You operate a jet engine at very low oil level, say 1 QT. Would oil pressure increase noticeably?
The value, 1 qt, is irrelevant without knowing the specific aircraft engine in question. The measurement is also irrelevant, as the engine may be measured in gallons, litres, pints, quarts, etc.
Would oil pressure increase with very little oil? Generally, no. Typically temperature will increase. However, oil pressure may increase, depending on the condition and temperatures involved. Larger volumes may cause a bypass to open, dropping pressure, whereas low volumes may not have that capability, even under pressure.
The type of cooling system, place the temperature and pressure is measured, etc, also vary the answer to this question.
How can oil pressure increase if the oil is low? Of course, the oil pressure will decrease and reach 0 when the oil supply will reach 0.
Not necessarily. I've had zero oil with pressure still indicated, and yes, it can happen. This is one reason why all parameters must be considered, including temperature, and ancillary or supported accessories such as CSD temps, engine operation, annunciators, chip detectors, etc, to determine what is really being seen. Conversely, it's quite possible in some systems to get a zero oil pressure indication with a fully functioning, full oil system.
As stated previously, some engines will run a long time with no oil, and may indicate accordingly.
The oil temp is not the critical engine parameter on Jet engines so you should not worry about it too much. The critical parameter on jets is the ITT (Inter Turbine Temp).
Oil temperature is a very critical parameter on a turbojet aircraft, whereas some of the most critical components use high pressure oil jets for lubrication and cooling: the turbine bearings.
Many turbojet engines don't have ITT. Other temperature measurements such as EGT and TiT are also used. Again, you must specify the specific powerplant.
Oil temperature on a turbojet engine can be extremely critical: a high temp on one powerplant, for example, may indicate an impending CSD failure. A CSD failure can lead to an uncontrollable engine fire in some cases...a magnesium fire you can't put out. Oil temp may be your first, or even your only warning before that failure occurs.
On pistons the critical engine parameter is the oil pressure. If the oil supply is low the oil pressure will be low and the oil temp may increase
Numerous parameters may be said to be critical on a piston engine, but then again, it really depends on the engine and which particular parameter it is to which you're referring.
Oil pressure does not decrease in proportion to oil level in most piston engines, nor in most turbine engines, either. If that were the case, we would be able to tell the oil level by the oil pressure at any given time. With a very low oil supply, so long as the temperature has not increased significantly enough to drop oil density and pressure, the oil pressure will remain a general regulated value until such an insufficient quantity exists that the pump cavitates. At that stage, temperature will generally be a good indication of the oil loss, provided the loss is not catastrophic in nature.