You seem to have missed a critical part of sneak attack, and you seem to not understand the mechanics of grappling in much detail.
I'll deal with Sneak Attack first. I DM for a group that includes a rogue. He has been able to sneak attack about 95 % of his turns in combat so far, for one simple reason: the group has a tank that stands next to the enemies. The important part of Sneak Attack is as follows:
You don't need advantage on the attack roll if another enemy of the target is within 5 feet of it, that enemy isn't incapacitated, and you don't have disadvantage on the attack roll
If your group has a tank, they are most likely going to stand next to the enemies. Just make sure you're attacking the enemies next to your tank, and you sneak attacks are going to be available pretty much every turn.
Now, on to grappling.
Grappling can never activate sneak attack, since they are special melee attacks, not special melee weapon attacks. Grappling replaces one of your normal attacks when you initiate the grapple. Once you have grappled the enemy, you don't have to commit anything on your turn to keep the grapple going. There are two ways a grapple can end: If you intentionally let go of the target, or if they try to break free. The first option requires that you actually say that you let go of the enemy. The second option forces you to do an opposed athletics check, but on their turn, using their action. Once you have grappled your target, you can use your action on subsequent turns to attack them. If you have the grappler feat, or if an ally is within 5 feet of the enemy (likely the tank of your group, if you do a bit of team coordination), you will be able to sneak attack them.
Your build seems to be trying to solve a problem that, for the most part, doesn't exist if you play in a group. Simply coordinating with your tank so you're attacking enemies within 5 feet of them will activate your sneak attack every time, unless you have disadvantage. Rogues in 5e are balanced around being able to sneak attack pretty much every turn. If they can't, they fall behind on damage compared to even the most tank-focused of fighters.
The barbarian's rage works, and can be a powerful combination. The monk's martial arts is controversial, depending on whether your DM counts the animal attacks to be "unarmed strikes" or not. For both of those classes, you'll also need to work out with your DM how Armor Class calculates out.
Rogue Sneak Attack will rarely work, if ever, as animal attacks are not finesse weapons. If you choose an animal form that bases its attacks on dexterity, a generous DM might call that a "finesse" attack and let it work, but that's not RAW.
Also note that to multiclass, you must have a 13 in the primary stat of both classes, which you do not have there. To multiclass druid with barbarian will require a 13 Wis and a 13 Str; for monk or rogue, you'll need 13 Wis and 13 Dx. If the numbers you show are before racial bonuses, then you can make it work. Otherwise, you may have to wait until 4th level in your starting class, to get an ASI and raise your stats enough to change to the other class.
Best Answer
There is no RAW way to combine Sneak Attack with Unarmed Strikes
In order to allow this, you'd need to either waive the Sneak Attack requirement for a Finesse weapon, or create a homebrew item that work as a finesse weapon but still an unarmed strike.
Even Unearthed Arcana has the major warning that it is not tuned for multiclassing, but once you start multiclassing with UA or Homebrew, you enter unprepared territory. It's up to the DM and they're table to decide how they want things to work. Allowing certain interactions (i.e. if the spiked glove is considered a finesse weapon), may break certain other things (i.e. rogue sneak attack.) There are generally trade-offs for making decisions and if those tradeoffs disappear because of homebrew, then you may end up with a character who is more powerful than they should be.
The dangers of allowing this interaction
But in the end, it's up to the DM decide if those interactions break their game. Sneak attack was purposefully made to NOT integrate unarmed strikes. If you have a way to turn an unarmed strike into an armed strike, it may cause a problem. If you don't think it does, or you can handle it by increasing enemy difficulty, etc, then go for it.
A note on unarmed strikes(from guildsbounty and Miatog)
It is important to note that monk unarmed strikes are not necessarily nor are they always a punch. The strike can come from an elbow, knee, head, etc - which is why the Monk doesn't need a free hand to make use of their unarmed strikes.
In addition, once you add a finesse weapon to allow Sneak Attack, you are no longer making an unarmed strike. You have armed yourself with a finesse weapon.