A black dome is functionally identical to darkness*, with a reduced radius. Objects passing through an illusion reveals that the illusory object are not solid. In most cases this is sufficient evidence to believe it to be illusory since an illusion is generally of a solid object. Since magical darkness is also intangible, there is no reason to believe it is an illusion merely because an object passes through it. Without this major indicator revealing the true nature of the illusion, it incredibly difficult for a casual observer to identify the illusion as such, and therefore takes an action to investigate it. Nothing in silent image's description suggests the spell ends when something touches it.
By that reasoning no amount of investigation can reveal that it is an illusion. Or really, reveal that anything is an illusion, because it could just be some other magical effect.
The way I read the spell description, the first sentence suggests a way to raise doubts regarding an illusion (oh! this door isn't solid). The second is the method to verify those doubts (I think I'll take a closer look at this door to figure out why my hand passed through it). The third is the result of succeeding on the check (it's an illusory door!). If merely touching the illusion was sufficient then anyone could use their environmental interaction to poke the potential illusion instead taking an action to examine it (and bypass any check, too!). It seems odd to me that a spell that outlines an action cost and ability check could be bypassed accidentally.
Physical interaction prompts a check. Firing something through it is interacting with it physically.
This is purely my own opinion on the spell, though hopefully it now makes sense why I rule the way I do. I tend to favour anything that helps players rely on interesting ideas rather than endlessly attacking the problem until it goes away, and ruling generously with illusions is one way to do that.
I suppose I don't find it to be an interesting use of illusion. Using a 1st level spell as a superior version of a 2nd level spell is rules abuse at best. Thankfully, it doesn't work by the rules so it is easy to fix.
The intent of an illusion is to deceive, not duplicate magical effects of higher level spells at will.
*I have seen some debate over the nature of magical darkness, though the majority tend to agree that it creates a dark bubble that can't been seen through. We operate according to this at my table because arguing about how a spell affects EM radiation is less fun than adventuring.
A good rule of thumb to use to figure out whether the rules are being correctly interpreted or working as intended is to think about whether the players would be upset if they were applied against them.
If the enemies used this tactic would the players be okay with it? Or would they feel frustrated and cheated?
Think about how it would affect the game world too. Illusion would be very powerful.
But this is all off topic.
On topic I definitely wouldn't use an attack cantrip as a high elf if my class has other attack cantrips with my main stat.
Vicious Mockery is very good from levels 1-4. After that it suffers from enemies having multiple attacks and a poor increase of damage. Still, you probably have better things to do than Firebolt at level 5 too.
