In one of the early episodes of TOS don't we see a tellarite murdered by an andorian.
You are thinking of "Journey to Babel." And the Tellarite was murdered by an Orion disguised as an Andorian.
If this kind of stuff is going on, I would hardly think that both planets would be founding members in a federation based on peace.
1. "Journey to Babel" established that the political situation leading up to the Babel Conference was extremely unusual. The Federation was on the verge of civil war over the issue of admitting Coridan as a Federation Member because there were so few people on Coridan that non-Coridanites were coming in and very cheaply mining its dilithium, to the point where Tellar was even claiming Coridan as part of its own territory (the same way the State of Ohio and the Territory of Michigan were both claiming ownership of the City of Toledo and its surrounding land during the Toledo War before Michigan was admitted into the United States as a state). So we can't take the situation in "Journey to Babel" as indicative of what Tellarite-Andorian relations were normally like.
2. That one Federate murders another, even if they are both high-ranking people, does not equate to their respective worlds not usually getting along.
3. You would never think in an era where Virginians were attacking Marylanders
en masse that Virginia and Maryland were both founding members of a Union based on peace, but they were. And they stopped firing on one-another after a few years, to boot. It's certainly possible that the Federation went through a period of internal tensions (presumably over the question of how much power the Federation should have over what issues vs. how much power its Members should have over what issues) that later abated.
4. As noted above, it was an Orion agent who was trying to incite a Federation civil war who did that, not an Andorian.
As far as symbolism goes... all flags, banners, emblems have a symbolic meaning, in every culture since time began.
Untrue. Not every aspect of every flag or banner has a specific meaning. There is no evidence that the colors of the Flag of the United States, for instance, were specifically chosen for any particular symbolism or meaning; any form of meaning given to them has been meaning assigned after the fact, not during it.
As such, it's entirely possible that the presence of thirteen stars in the TOS banner is actually arbitrary -- merely done because the designer thought it looked good and for no other reason.
A better question, I think:
Does the Flag of the Federation seen in TNG-era
Trek always contain the legend "UNITED FEDERATION OF PLANETS?" Is it possible that there is an official version of the Flag with the Federation's full name written underneath the emblem in Vulcan, Andorian, and Tellarite, for instance, and that they're all equally official?