1) A sg of 1.025 - 1.026 is ideal for corals and inverts. 1.027 isn't high enough to kill anything.
2) 20ppm nitrates isn't lethal to anything but the most delicate sps corals. Fish are the least affected, being able to tolerate 50ppm+.
Had your sg been at 1.024 and then risen, that could've killed the shrimp as they're very sensitive to changes in parameters. You also had measurable ammonia which may have damaged the clown's gills.
As for lighting for the nem, if you want something inexpensive that you can control with a $5 timer from Walmart, check out the RapidLED PAR 38 bulb. One should be plenty for a 5.5g cube.
BTW, the nem is looking better, but definitely needs more light.