Single-transistor BJT amplifiers won't come up trumps for all the "rules" you have written at any frequency. For instance, you require an amplifier so I'm assuming you want voltage amplification. OK so far?
The collector delivers a current output signal and not a voltage signal. This means that this current signal only becomes a sensible voltage signal when fed through a load resistor i.e. the collector resistor. This means that the output impedance of a common emitter amplifier is largely dictated by the collector resistor and if you want a small output impedance then you must use a small collector resistor.
A small collector resistor also means smaller gain so, you have a quandary - rule 2 is tending to fight against the word "amplifier" but let's say you can have a 50 ohm resistor in the collector for now and look at the implications.
If you have 50 ohms in the collector then you have an output impedance that is 50 ohms but to get gain you'll need a small value emitter resistor. Let's say you go for a 5 ohms resistor in the emitter - this will give you a gain of ten (you may well be wondering how I jumped to that conclusion of course). Here's why - the AC signal voltage on the base drives through the forward biased base-emitter junction therefore that input base AC signal is also largely seen on the emitter. The current thru collector and emitter are nearly the same hence if the collector-emitter resistor ratio is 10, it's a fair bet that the circuit amplifies by approximately ten.
This leads to another contradiction of the rules you specified - the input impedance is going to be limited to just a few hundred kohms with an emitter resistor of 5 ohms - basically you can make a rough calculation of input impedance by using Hfe and emitter resistor - if Hfe is 100 and emitter resistor is 5 ohms then input impedance is 500 ohms - it's a decent approximation method not an exact method so don't hang me on this.
Given also that your biasing resistor rule implies 10% of colelctor current flowing thru them your input impedance is not high - it's a messy-low-to-middling couple of hundred ohms.
Now that I've assasinated your rules, please consider that a two transistor solution with a common emitter followed by a common collector stage is much more practical to the rules you have written and, for RF amplifiers a common base tolology is quite often implemented followed by a common collector circuit.