I'm working on a set with a vanilla matters theme (set in dino-land no less), and I'm curious how the following interaction works:

Muraganda Petroglyphs + Grizzly Bears + Savage Summoning = ?

Am I summoning a 5/5 Bear (2/2 Bear + Petroglyphs + +1/+1 counter) because Savage Summoning says the card may be cast as though it had Flash without actually temporarily hard-coding "Flash" onto my bears?

Am I summoning a 3/3 Bear (2/2 Bear + +1/+1 counter) because my Grizzly Bears became an Ashcoat Bear for this turn?

5/5 Bear for the reasons you stated.

Neither the bear spell nor the bear permanent it becomes gain flash or any other abilities. Savage Summoning just grants additional permission to cast the spell. So the bear permanent is 5/5.

Philosophers all around the world will get sad at this.

The followup would then be:

Some random card
Creature - Dude
Creatures with no abilities that you control deal combat damage as though they had first strike.
"I'm an example!" - Dude
2/2

Would this cause my bear to lose his +2/+2 or not?

Your bear would not lose the bonus.

This card just stating something about the your creatures, setting up a special rule for them during combat. It is not giving an ability to them.

Sweet, thanks.

Rule of thumb: if the effect says your creature(s) "has/have" or "gain(s)" something, it is giving an ability to them (and therefore they lose). Otherwise, it isn't.

As always, you need to look at the Oracle text of the card - due to some common effects becoming keywords, things like Stamina, Goblin War Drums or Shallow Grave end up receiving errata to grant abilities.

Note that you need to say "assign" rather than "deal" in that sentence to get it to do what you want.