I finally took my newly built bike out for a test spin, about 9 miles. Coming from my carbon roadbike, I am shocked at how much the fork flexes when the front brake is applied especially considering I'm a short guy and only weigh 120 lbs. I've disassembled the headset twice and re-adjusted/torqued everything and there is no play in the headset. You can even see/feel the flex when gently rocking it back and forth while applying the brake.

This has me a bit freaked out because it I've never experienced this, even as a kid when I used to ride cheap steel BMX bikes. So I have a few questions, hope y'all can help.

1. Will a headset with sealed/cartridge bearings lessen the flex?
2. Even if it's normal, I'm worried about the fatigue on the fork/steerer tube. Can I eliminate the flex by using a carbon fork? If so, does it matter if the steerer tube is alloy or carbon?

FYI it's a new steel frameset with a new (but cheap) Neco headset

The apparent "flex" may be movement between the cheap headset cups and the headtube. It could also be the cheap headset compression ring not doing a good job maintaining a tight fit between the fork steerer tube and top bearings. About the only solution is to install a better quality headset.

So it's actually movement or slop between components then, and not actual flexing of the legs themselves?

The fork legs themselves will flex somewhat depending on the tubing size and thickness, but certainly not to the point of being a problem. Looking at the photo in the OP, the fork appears to be a rather average steel unicrown design that won't be any more flexible than other similar forks.

I think it's reasonable to replace the Headset with a quality sealed unit as I'd have to do that even if I decide to upgrade my fork later on down the line. I'd also like to hang on to this bike for a while. Any suggestions for a nice headset (1 1/8")? Looking at some FSA's.

FSA makes excellent headsets at reasonable prices. I've got an Orbit X on my Rush, and it's as good as any I've seen. FSA Orbit X headset | Retrogression Cane Creek is very good as well.

I have the same bike. I got this headset

Ritchey Logic Threadless Headset 1 1 8 Black | eBay

I replaced the plastic top cap with aluminum but the headset itself it great bang for the buck

The 40 series is a great value.