Did Vito not fully trust Tessio and Clemenza?
When they first started out, he split them off, one in the Bronx and one in Manhattan.

And if they were old school friends, why would he mandate to them not to see each other through the years?

I'm not sure about the reference to school friends?

He trusted them as much as he trusted anyone in that business. The Don was a very cautious man and wanted to ensure that only he (and perhaps Genco/one of the Don's sons) had the big picture about what was going on.

So this means that Clemenza doesn't need to know what orders the Don gives Tessio and vice versa. Also both men together control almost of the Family's soldiers so the Don didn't want them to be too chummy with each other or get ideas about making changes at the top.

yeah, not about trust...

alliances, loyalties aren't concrete so to protect your own interests.....you structure your organization in a way that minimizes the chances of a coup being staged...

Clemenza went to C.U.N.Y and Tessio went to Harvard. Tessio was always smarter.

As the novel states, Vito split them off to reduce their opportunity to plot against him. Since Tessio was smarter (and a Harvard graduate), he caught Vito's drift right away. Clemenza's only concern was with the availability of restaurants in the Bronx.

Also, the novel states two other reasons:
First, Clemenza was more impetuous than Tessio, and needed to be kept on a tighter leash. That's why Clem lived at the Mall, but Tess didn't.
Second, it was to Vito's advantage that his foes think that Tessio's distance meant that Tess was some kind of independent operator, the better to have the foes underestimate Vito.

In GF2 it seemed that young Clemenza at first underestimated Vito, perhaps because Vito was just a neighbor who had done him a favor (hiding the guns). In the spaghetti scene he seemed more skeptical than Tessio. However, later he's pretty humble, for example in the deleted scene where he introduces young Hyman Roth.

Yes, the turnaround was remarkable. In an earlier deleted scene, following the dress factory robbery, Clem was still lording it over Vito--barking orders about where to make deliveries, making him wait while he did the job on one of his female prospects. But in the scene you cited (after Fanucci's murder), Clem's atttitude turned around completely. Nice touch: he takes off his hat in Vito's presence.

I wonder if Clem's change from being such a hard ass to being somewhat meek demonstrates that maybe he knew his limitations and put on a tough front to fool strangers into thinking he was tougher than he was. After all he was pretty panicked when he stupidly threw guns to Vito (a complete stranger) and then tried to pretend the rug he was stealing was in a friend's house.

That's a likely scenario, dt. Vito really pulled off a coup pretending to be passive (even meek) and letting Clem brag and show off. But after Clem and Tess showed they didn't have the coglione to challenge Fanucci, Vito stepped in with a perfect, gutsy solution. Clem's bigshot days were over. What I like about Clem and Tess at that point was that they were smart enough to recognize Vito as their superior and go along with his leadership, instead of resenting him and working against him. That took practical smarts.

It was also a matter of pragmatism. Everyone was under Fanucci's thimb, and then Vito defeated Fanucci. It was clearly in their best interests to be aligned with him.

I love the dynamics of that scene.

While it's obviously the first time we see a hint of what Vito would become, you also see the differences between Tessio and Clemenza, and that Tessio was always smarter.

Tessio is the first one to mention the idea of not paying Fanucci.

Clemenza isn't inclined to listen to Vito at all, and tries to dominate the conversation with his ideas, while (other than saying that someone else must be collecting for Maranzalla) Tessio says nothing at all.

He clearly is considering what Vito has to say, and immediately agrees once the plan is laid out.

Clemenza pouts and then reluctantly toasts, either upset that he's being usurped or lacking the vision to see that Vito has the intelligence and force of will to make his plan work.

But most importantly: the also liked Vito.

Yes, but at the end of GF, Tessio, turned traitor, said: "Tell Mike it was business. I always liked him."

I think Clemnza liked Vito but I am not so sure about Tessio. Clemenza was an extrovert who enjoyed his food, his wine, his dancing and his wife and kids.
Tessio, the smarter one was purely a businessman looking out for his own best interests. As soon as he thought the takeover by Michael was a done deal, he figured the Corleones would be toast and he turned traitor. I don't think that thought ever crossed Clemenza's mind.

Fair enough.