Can I use this remote sensing circuit for measuring current measurement (10uA to 1A) with series resistors 10K, 100R and 1R operated by relays?
Sure that will work. The sense resistors could even be switched by power MOSFETs. The remote sense will remove the voltage drop from the sense resistors and any switch.
But like I said, if you want an output which can go both positive and negative, a different output stage will be required than a simple voltage regulator. It will resemble an audio power amplifier output stage.
If you want to measure currents that low, then some attention will need to be paid to make the remote sense circuit not draw too much current itself but that is not difficult. Remove sense bias currents from nanoamps to picoamps are easily feasible using operational amplifiers as buffers.