You cannot use the wand more than once per turn with these two features

Using a wand is a special action defined by the wand's description. The prototype wand says:

As an action, you can cause the magic wand to produce the cantrip [...]

You cannot produce a cantrip from a wand with the Attack action.

When you take the Attack action, Arcane Armament allows you to attack two times in that one action. Attack meaning to attempt to hit something with a weapon. But you cannot replace an attack with another action. And using the wand to produce a cantrip is not considered, in and of itself, to be an attack. Cantrips may have attacks as part of their effects, but producing a cantrip is not itself an attack.

When you produce a cantrip from the wand you have to use your full action to do so, which means normally giving up the option to Attack.

So, you cannot take an attack given to you by the Attack action and use it to use the wand to produce a cantrip.

Yes, artificers can be counterspelled

Although the "flavoring" is different, artificers' spells have the same kinds of components that other spellcasters do. In essence, rather than drawing on some innate magic or some eldritch power, or manipulating the Weave through studied techniques or through music, artificers instead cast spells by infusing magic into items and then releasing it from those items.

This distinction is described in more detail in the sidebar "The Magic of Artifice" on p. 4-5 of the UA:

As an artificer, you use tools when you cast your spells. When describing your spellcasting, think about how you’re using a tool to perform the spell effect. If you cast cure wounds using alchemist’s supplies, you could be quickly producing a salve. If you cast it using tinker’s tools, you might have a miniature mechanical spider that binds wounds. When you cast poison spray, you could fling foul chemicals or use a wand that spits venom. The effect of the spell is the same as for a spellcaster of any other class, but your method of spellcasting is special.

The same principle applies when you prepare your spells. As an artificer, you don’t study a spellbook or pray to prepare your spells. Instead, you work with your tools and create the specialized items you’ll use to produce your effects. If you replace cure wounds with shocking grasp, you might be breaking down the device you used to heal and creating an offensive item in its place—perhaps a gauntlet that lets you channel a surge of energy.

Such details don’t limit you in any way or provide you with any benefit. You don’t have to justify how you’re using tools to cast a spell. But describing your spellcasting creatively is a fun way to distinguish yourself from other spellcasters.

As the last paragraph clearly states, artificers' form of spellcasting is unique in the way it is described and roleplayed, but mechanically it has just the same components as any other class' spellcasting. As such, other spellcasters can perceive artificers casting a spell, and can counterspell them.