Scinerio:
I have a Big Barda and Mr. miracle duo figure that is in play due to merge. The duo uses up BB's boomtube or even all three traits from BB and MM's cards. Next, the 2 figures split. Maybe solo BB uses the boomtube now, but eventually they merge again.( Now here's the main question.) Can the BB/MM duo now use BB's boomtube again because they are once again new to play?

No. They aren't new. They are the same characters and any effects which were previously in play (maybe the Duo had a relic attached?) would carry over to one of the single characters of your choice.

What if you actually have a second duo figure on your sideline? Would it then become a new game element.

Nope. When BB and MM merge and split then merge again they carry all their baggage.

It doesn't matter which physical piece you use, what matters is that they are still a replacement character, so they are considered the same character that Big Barda was.

It works, kind of, but not how I think you want it to.

If you have BB and MM (split up), and you use Boom tube invasion, BB will have the game effect [used BTI this game]

When you merge them, the merged figure will also have [Used BTI this game]. So the merged figure will not be able to use BTI (because it would violate the 1/game clause)

When you split the duo up, you get to assign who gets what game effects. So the trick is to give MM the [used BTI this game].

Now when you go to try and use BB's BTI, she checks for a game effect saying she has used it, and doesn't see it there, so she gets to use it again.

Then you rinse and repeat, making sure to always put all of the [used BTI] on MM.
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If you are trying to use BTI with the duo more than once a game, you can do it, but you need a third character (a second miracle is probably best).

The chain would look like this.

(Duo= Big barda and mister miracle duo, BB= big barda, M1= miracle that starts on the map, M2= miracle that starts on sideline)

1. Duo uses BTI
2. Duo splits into M2 and BB, putting [Used BTI] on M2
3. BB merges with M1
4. Duo uses BTI
5. Duo splits putting [used BTI] on BB
6. BB merges with M2
7. Duo splits, putting ALL instances of [Used BTI] on M2
8. Repeat steps 3-7 ad-infinitum.

Everyone wants the duo to do it multiple times for some reason, no one cares if Barda can.

Okay, I follow. My only question is how can you put a used BTI on Mister Miracle when he doesn't possess the BTI trait?

Split says "put the effects wherever you want"

it says nothing about whether it makes any sense for them to have it or not.

There is nothing suggesting that you have to put a game effect somewhere it might have shown up without the use of split.

So, if you split up a War Machine and Iron man duo into war machine and silver centurion, you can put any [has used PBT] effects on war machine, etc.

The fact that he doesn't have PBT isn't a reason you can't do it.