It activates only once
The trait reads (emphasis mine):
When you damage a creature with an attack or a spell [...]
Notice how "attack" is written in lowercase. This indicates a single attack. Not an Attack Action. When you have extra attacks, you attack multiple times with one Attack Action. But when you activate Fury of the Small for example on your first attack of your multiple-attack Attack-Action:
you can't use it again until you finish a short or long rest.
This means you can only use this effect on one of your attacks. The same applies to spells.
Except when you use AoE spells
In the comments Erik raised a very interesting point for this discussion:
Can you elaborate the difference (if any) between a multi-attack spell like Eldritch Blast vs an area spell like Fireball? (The former makes multiple attack rolls, the latter only has one damage roll which is used against every target) ?
The difference is that an area spell is still one single spell and that you don't make a spell attack. The trait does not limit the effect to spell attacks, but instead to spells that cause damage. Instead the targets make saving throws. That means that if you use this trait with an area spell like Fireball you should be able to add the extra damage to every single target that takes damage from your spell. Note that some AoE spells also have a clause that mentions a target taking damage even if they make their saving throws. These targets would still be valid targets for the additional damage.
If you instead use the Eldritch Blast you have the same situation as I already described in the case of multiple-attack Attack-Actions: you attack multiple times with a spell attack. This means you can only add the damage to one of these attacks.
Personal Addendum
I feel like the described AoE damage increase is not really the intended use of the trait, but according to the Rules as Written you would be able to add the extra damage to any target that takes damage because of your AoE spell. As a DM I might rule that the trait should read:
When you damage a creature with a [weapon] attack or a spell [attack] [...]
This would disallow AoE spells and other spells that use saving throws. It would only apply to real attacks against single targets and could only be used once.
Another way of changing the trait might be:
When you damage a [single] creature with an attack or a spell [...]
This would make it clear that the damage has to be focused on a single creature. In general this would disallow AoE spells, but then again it may be the case that it just so happens that only one enemy was caught in your Fireball, which would allow this again. While the second version might be interesting for some people as a way of changing the trait to deal with the problem of applying the damage to multiple enemies, I would recommend to use the first "fix" to simply disallow AoE spells.
Yes, there is now an (un)official statement
Jeremy Crawford has clarified in a Tweet:
Q: A firbolg PC casts booming blade then uses hidden step to go invisible. The creature that is hit with booming blade moves away and
thus takes damage. Does hidden step deactivate?
A: Hidden Step ends if you make a damage roll, no matter when you make
it.
Q: So it doesn't matter if a spell/effect is initiated by you before
hidden step was activated? As long as you make a creature do a saving
throw or attack or roll damage for any reason then hidden step ends?
A: That's correct.
So, passive or not, as long as a creature is making a saving throw or taking damage from an effect you initiated (no matter when you initiated it) for any reason, hidden step will end.
This matches the intent we can understand when reading the ability itself.
Intent is clear from reading as well: if the user causes a harmful effect, hidden step deactivates.
The ability is explicitly written to restrict just about every way you can damage or harm an opponent (regardless of when said effect was initiated), so it seems clear that the intent was to prevent the user from doing harmful things. As such, allowing a loophole for spells and effects cast/initiated before the user used hidden step runs counter to that intent.
It may also be helpful to note that the designers also intentionally wrote this ability so as not to restrict beneficial spells and abilities being used.
The ability is already quite powerful and from a DM standpoint this argument would seem like an attempt to cheese additional power out of the ability by circumventing its primary restrictions.
This also seems wholly fair. In the example of booming blade, the only reason to cast such a spell is to try to cause damage. If you also choose to use hidden step you are taking the risk that the opponent triggers that damage and thus cancels your ability. It was player choice to cast the spell and to use the ability.
Allowing this opens the door for plenty of bad things/cheese
Our heroic firbolgs casts delayed blast fireball at a group of 4 enemies, uses hidden step, then immediately stops concentrating on the spell. It takes no action to stop concentrating on a spell so this is the exact same scenario as with booming blade. This causes 4 saving throws and 4 creatures to take up to 12d6 damage.
Does it seem to be intended to allow such an effect to not break hidden step? This seems like an overtly harmful action and I really do not think for one second that the designers intended this to be the case.
And this is only a cursory look at one spell. I'm sure that there are many more and possibly even worse spells/effects one could get away with through this loophole.
Neither RAW nor RAI support allowing a loophole for effects initiated before hidden step is activated.
As always, a DM may, of course, allow this if they want to for any reason.
Best Answer
Yes, you can take your next turn while still under the effect of Grovel, Cower, and Beg. It doesn't make a huge amount of difference, though; you can't benefit from your own groveling. Only your allies gain advantage, not you, so while you could do something on your turn that triggers readied attacks by your allies, which would then have advantage, that would be a pretty unusual set of circumstances.
As far as going prone, no, the action doesn't specify that you go prone to do it. Prone is a fairly specific position, and the condition applies a moderately serious penalty. I wouldn't apply that unless it's specifically called for or very obviously applies (such as if an attack comes when you had just laid down for a nap).
If you need arguments to present to your DM, the main thing I would note is that there's a significant difference between kneeling or crouching and being prone (i.e. flat on your back or front). Getting up from prone generally involves a lot of maneuvering to get your feet back under you, while if you take a knee, you can spring right back up and straight into motion at almost any time (depending on the condition of your knees). As generally understood, cowering doesn't mean planking or something -- you aren't stretching out full length on the ground, you're crouching and trying to be as small as possible.