No, even monsters only get one action per turn, and actions should not be confused with attacks. in the case of monsters that can attack multiple times, that's because Multiattack is one action, that just happens to result in attacking more than once. Each attack is not an attack action! Multiattack is also very specific about which attacks the action results in.
So an owlbear can indeed grapple or shove, but that grapple or shove is done by taking a normal Attack action, not a Multiattack action. The Attack action is general purpose and more flexible than Multiattack, but doesn't allow the owlbear to make more than one attack during the action.
Default play: gridless.
Nothing stops a grappled creature from shoving.
"Grappled" zeroes out your speed, and that's it. (PHB p.290, "Appendix A: Conditions")
However, this may not break the grapple.
As you've quoted, the effect ends if the grappler's reach is exceeded. On a successful shove you can either knock your target prone or push them 5' away (PHB pp.195-196, "Shoving a Creature").
Knocking the grappler prone doesn't--by itself--move you out of the grappler's reach.
Shoving the grappler 5' is... uncertain.
- If you contend (as I would) that while grappled there is a functional distance of 0', then shoving 5' extends the distance to 5' and doesn't exceed the grappler's reach: grapple not broken. See also When you grapple an enemy, where are they?.
- If you contend (as at least one or two others do) that there's no room in D&D for 0' separation, then a 5' shove would exceed the grappler's reach: grapple broken. Factors in favor of this interpretation: "whether a creature is a friend or an enemy you can't willingly end your move in its space" (PHB p.XX) plus the notion that one's "space" in gridless-world is larger than one's physical extent.
The one definite case I can think of (just for fun):
- You have the Charger feat.
- On your turn you Dash at least 10' into the 5' reach of an opponent...
- ... who had previously Readied a Grapple.
- Your GM rules that the Grapple (attempt) fires off before you can take your bonus-action Shove (from Charger).
- The grapple against you is successful. You're grappled.
- Now you get your shove, which is successful.
- Your grappler is shoved 10' away (per Charger), breaking the grapple.
Summary: this sounds like an excellent space for a ruling, leading to a house rule.
Variant play: grid.
Nothing stops a grappled creature from shoving.
same as above
This will break the grapple of a 5' reach.
In grid-world even when grappled creatures are at a 5' "distance" because, as @daze413 rightly points out, grappling doesn't move one into the same square.
The shove then moves the grappler 5' farther away, exceeding a 5' reach. So: yes, in grid-world the equation is
grappled + 5' shove = 10' separation
Best Answer
You are able to shove or grapple while prone.
The Basic Rules outline the prone condition:
Nothing here says you can't grapple or shove while prone.
Reading through the rules for grappling and shoving, there is nothing there either which would prevent you from grappling or shoving while prone.
Note: while the rules for prone say "The creature has disadvantage on attack rolls", grapple and shove do not use attack rolls! Instead, each of these say that "instead of making an attack roll", you make a Strength (Athletics) check. So you don't even have disadvantage to grapple or shove while prone!