I'm researching export controls on hypersonic missiles. I was inspired by this link, however I've run into few problems that I can't seem to find answers to:
1. Does the U.S have any plans to sell hypersonic missiles to other countries? If not since they probably don't now, do you think the U.S ever will or has the desire to sell to allies?
2. Do you think Russia and China follow-on is true if the U.S implements export controls? Will Russia and China also implement export controls if the U.S does?
3. Will hypersonic missiles be a substantial part of the arms market in the future? Or will this be a niche weapon that the US tries to control?
Thanks for you time and any help is appreciated!

So far as I know, the US has no plans to export the new breed of hypersonic missiles. (Existing ICBMs are, of course, hypersonic missiles, and we don't export those.) At present, the proposed hypersonic missiles are still in development, so there's nothing to export as yet in any case. But even when they are deployed, I can't imagine that they would be exported as they are likely to be considered cutting-edge weapons that give the US an advantage we wouldn't want to share. Now, maybe down the road, a decade or two from now, in a more hostile world, we might consider exporting them (or the technology to make them) to, say, the UK, but that's speculative.

Likewise, I can't imagine Russia or China sharing their cutting-edge hypersonics technology to anyone else.

From what I’ve read, I think it may be still too early days to know if the US plans to sell hypersonics. A recent NYT article states that “In 2018, Congress expressed its consensus in a law requiring that an American hypersonic weapon be operational by October 2022. This year, the Trump administration’s proposed defense budget included $2.6 billion for hypersonics, and national security industry experts project that the annual budget will reach $5 billion by the middle of the next decade. The immediate aim is to create two deployable systems within three years. Key funding is likely to be approved this summer.”
 
So to me, it doesn’t sound like the timeline and production scale would allow for sales in the near term and from a strategic perspective, I would be surprised if the US would want other countries to have these given the military advantage they offer. In some ways, they strike me as what it must have been like to acquire nuclear weapons when they were first developed, from the perspective of deterrence and power. To your other questions, I would normally say that the same types of logic and agreement that the nuclear powers operate on (“we’re allowed to have these weapons because we are responsible states, but others cannot obtain them”) could apply to hypersonics but given how poor relations are right now between these countries I’m not sure they’d all agree to be bound by export control limitations.
 
I’d encourage you reach out to others and keep reading. SIPRI is an extremely reliable source of information, and it has done some solid research in this area too:
 
https://www.sipri.org/sites/default/files/Factoring-Russia-into-US-Chinese-equation-hypersonic-glide-vehicles.pdf
 
https://sipri.org/commentary/topical-backgrounder/2017/chinas-calculus-hypersonic-glide
 
France is also reportedly interested in their acquisition: https://www.defencetalk.com/race-for-hypersonic-weapons-heats-up-as-france-joins-fray-72093/

There have been a couple related articles you might want to check out, if you've not seen.

https://www.nytimes.com/2019/06/19/magazine/hypersonic-missiles.html 

https://www.armscontrol.org/act/2019-06/features/arms-race-speed-hypersonic-weapons-changing-calculus-battle

I was recently looking through the topic wording and technology regarding missile defense peaked my interest. I was told by another member of the community that ballistic missile defense technology was not categorized as "arms." 

In my brief reading through topic literature, I cannot find anything that speaks in a particular direction. It appears to me that most BMD systems use surface to air missiles, which I would think fall under category four on the USML. However, I cannot find any explicit reference to these operating systems being classified as arms under US code.

The controlling document, at least as explained to me, the AECA, was written well before the explosion of missile defense technologies during the 80s and afterward. I was wondering if you could point me toward any definitions of what constitutes arms sales or sales of arms that definitively create a distinction that allows missile defense to either be considered or not considered a sale of arms.

Thanks for your query. Alas, you've been misinformed, at least to some extent. Ballistic missile defense systems certainly are considered as arms and you'll see many sales falling under the foreign military sales (FMS) and direct commercial sales (DCS) programs. To date, the largest single FMS notification under the Trump administration has been for Terminal High Altitude Area Defense (THAAD) equipment, support, and services to Saudi Arabia. See the Forum's resource page and google spreadsheet for many other missile defense sales.

You can also find some more discussion of specific missile defense sales from the United States and more globally in this recent publications:

William Hartung and Christina Arabia, "Trends in Major U.S. Arms Sales in 2018: The Trump Record - Rhetoric Versus Reality," Center for International Policy, April 2019.

Aude Fleurant +, "Trends in International Arms Transfers, 2018," SIPRI, March 2019. (see box 1)