Switching a ~10 μA current at ~10 kHz
I'll start by answering your questions directly, then giving some thoughts you can use depending on your actual goal

(1) No, this is not a reasonable approach. Let's look at the period of your frequency at 10kHz -> 1/10,000Hz = .0001s, or .1ms. This means that your circuit needs to go from off, to on, to back off in .1ms. 

Looking at a typical small-signal relay, such as the one below, the rated "on" time is 7ms, and the "off" time is 3ms. Meaning that if the circuit were being switched by this relay, the fastest it could reasonably go is about 100Hz (10ms = .010s, 1/.010 = 100)
http://www.mouser.com/ds/2/307/en-g5v_2-536510.pdf

At the end of the day, a relay is something that takes an input and switches something else, and so is a transistor. However, a transistor is much faster and more efficient - it just can't do some of the things a relay can. You're better off just using a transistor. 

Neglecting the relay, just start off thinking of your transistor as a perfect switch, then look up the specs (on-voltage, etc.) and transistor formulas that make it not perfect, and get a better idea of how your circuit will behave. What's the time constant of your RC-based load? Can it respond in .05ms? Look up the datasheet on the 2N3904, if you substitute in your formulas for a BJT, will it work? What current will flow through it, and thus your load?

After you look at how much current is going through/ into your BJT, is it reasonable? Will the transistor need a resistor on the base to protect it?