Learn English – “There should be someone at the reception desk”

Should in this context means in case or if. It's a more formal way of referring to a hypothesis.

So the meaning of your quotation is: Some of the strangest and, in case they prove correct, most revealing insight of modern physics.

Should you need more information on this structure, don't hesitate to ask (=In case you need...).

I think here we go is more commonly used to indicate the beginning of something.

By contrast, there we go is more commonly used to indicate the completion of something.

For instance, here we go can be used when going somewhere.

Here we go on another adventure to the movies!

Here we go can also be used more negatively, almost sarcastically.

"You don't want to travel for the holidays because you don't like my mother!"

"Oh, here we go! I don't dislike you mother, their guest room just smells funny."

In the above scenario, the second speaker believes a dispute is about to unfold, which is rudely indicated by their saying here we go.

For there we go, we commonly see it at the conclusion of an action. For instance:

Grandpa was very old, and had bad knees. It was difficult for him to sit on our low couch. i heard him strain with effort as he sat. "There we go" he said, with a sigh, as he finally settled onto the squishy pillows. "They don't make them like they used to."

There we go can also be used to express approval. For instance, if you're teaching a child to ride a bicycle, when they do a good job, you could say, "There we go!" In this situation, you could also use "There you go!" However i think because the activity is together, either is appropriate.

Similarly, it could express approval of more removed events. For instance, your football team scores on TV, and you exclaim, "There we go!"

There may be other uses of these phrases, but I consider these the main ways to use them, which are actually fairly distinct, I think.

You also mentioned let's go. I think that can be used mainly to indicate that it is time to go. For instance:

"Is this a good time to go to dinner?"

"Yeah, let's go!"

It can also occasionally be used as encouragement.

For instance, as a sporting event, you can express support for your team, the Cubbies, by shouting "Let's go Cubbies!"