Yes, you can move before taking the bonus action attack
No timing is specified
On your turn, when you score a critical hit with a melee weapon or reduce a creature to 0 hit points with one, you can make one melee weapon attack as a bonus action.
The phrasing here says nothing about the precise timing of the bonus action. It can be simply read as saying that after the condition is met (critical hit or reducing to 0 hp) that you are granted the ability to make a bonus action attack. It never says the bonus action must be taken "immediately" or anything along those lines so the timing of the bonus action attack is not restricted at all.
You choose when to take a bonus action during your turn, unless the bonus action’s timing is specified
The rules for when you can take a bonus action say that if the timing is not specified, then you can choose when to take it. Thus, there is no reason why you would not be able to choose to move before taking the bonus action attack.
The rules explicitly allow moving between attacks
If you take an action that includes more than one weapon attack, you
can break up your movement even further by moving between those
attacks. (PHB 190)
Even within a single Attack Action, the rules allow one to move between attacks. It absolutely makes sense that they would allow it between an Attack and a bonus action attack for example. After all, even if the bonus action isn't considered to be part of the Attack action (or whatever triggered the GWM bonus attack condition), it is indeed a weapon attack and thus moving between them seems like an incredibly reasonable interpretation.
Jeremy Crawford has agreed this is the intended interpretation
Jeremy Crawford has clarified this exact issue via Twitter and agrees with the interpretation above:
The intent is that you could move before taking the bonus action in the Great Weapon Master feat.
Yes, this works
You have quoted all of the relevant bits, but the Animated Shield works the same as a wielded shield without Wielding it with your actual hands that are holding onto a Heavy Weapon.
This is also a Very Rare item - it's special and powerful!
Sage Advice confirms
The key phrase comes from the Animated Shield, which states, emphasis mine (DMG, 151) :
...protects you as if you are wielding it
"As If you are wielding it" is equivalent to "wielding it".
This interpretation is supported by Jeremy Crawford's Sage Advice.
The text of animated shield says the item protects you as if you were wielding it. To Shield Master and the like, you're wielding it.
Not without a cost
The investment cost is two of your ASIs (and you still need to get, and keep, that Animated Shield.)
(Don't?) Hold on a minute...
Just a reminder, but the Animation requires a Bonus Action and only lasts 1 minute. You only get ONE bonus action/turn so your first round would be used doing this (so no, extra attack for a crit/kill from GWM or any other Bonus Actions you may have available.) In addition, if your encounter lasts longer than 1 minute, the shield will fall unless you use another Bonus Action to start it back up.
Best Answer
The +10 damage bonus does not cascade, but it does apply to each attack.
While there are often time/speed benefits to rolling attacks in groups1 (some people like to use different colored dice), you still consider each attack separately. An 11th level fighter using Extra Attack will roll three separate D20s, evaluating hit or miss for each of them. After each hit, roll damage - any attack rolls you took the -5 penalty on get the +10 damage.
Compare the Great Weapon Master Feat's benefit...
...to a Barbarian's Reckless Attack feature:
Note the emphasized text in each. Great Weapon Master is a choice made on each attack, while Reckless Attack is made before the first attack and applies to the whole turn.
So, you can turn GWM on or off for each attack. Say you roll a 18 on the die, then factor in whatever bonuses (ability score, proficiency, etc.) apply alongside the -5 from Great Weapon Master... and the DM tells you missed. You know that 18 is a good roll by itself, which means that the target has a very high AC. On further attacks in the same turn, you could choose not to suffer the penalty (and not to gain the damage) to increase your chances of hitting at all.
1If you're going to roll all the attacks ahead of time for speed's sake, for fairness sake, you should declare if you're using GWM or not ahead of time. That way, you're not making a decision on information you don't actually have.