I currently have 2 rules questions I need help with. I think my group has been playing these incorrectly and I am unclear.

1) Can a flyer shoot over a grounded character on the same elevation? Adjacent and non adjacent.

We have been shooting over non adjacent characters and not shooting over adjacent close combat characters.

2) Lantern Constructs: how can you assign a lantern construct to a character and make them friendly to a lantern keyword theme team when certain lantern characters have a construct trait on their card required to have the constructs?

I have seen players just assign orange constructs with characters and using the lantern like they are inherent with orange lantern keyword.

Can someone please help me Clarify this?

1. No. Flight has no effect on a characters ability to draw lines of fire. It only affects movement.

2. The only way to give constructs to characters without a trait that allows it is with one of the Power Battery resources.

I am not sure I get your second question but if I am wrong let me know

Are you asking if you can assign a different coloured construct to one of the Green Lanterns that have the slot in their base?

i.e.,
If so, yes. As odd as it is, there is no restriction on what colour constructs can be attached to the bases of GL's who can use this ability

Other than those select characters and characters assigned a construct via a battery, there is , currently, no other way to assign a construct to a character

Also, the above demonstrates why it is a really good idea to quote the power in question or provide a more detailed explanation as to the source of the quandary

Edit: VP is right on the first question but, you know, [opinion]that is a really cool idea, having flyers being able to draw LoF over characters, by, in effect, kind of soaring up. I never liked the original soaring mechanic - it lacked something - and flyers seem to be missing... something.[/opinion] That said, it does not change the answer

As BigSoph mentioned, VP is exactly right on the first question and it's a good idea to quote the powers to make sure everyone understands what you are asking. I point that out because I had a different take on what you are asking so I want to make sure everyone is on the same page.

So regardless of the keyword(s) that a character has when they have a Power Battery on their force, they will gain the colored Lantern keyword of the ring that is attached to the Battery. If a player has 3 Green Lanterns (each with the Construct trait), the Green Battery, and a figure without the Green Lantern keyword, that other figure can gain the GL keyword by being assigned a Construct from that Battery.
To note, that won't make them a themed team (as that has to be determined during the building of the team and the Construct can only be assigned after the game starts)... that was one thing I was reading in your question that may not be the situation.

None of the Power Batteries require that the figure that a Construct is assigned to already BE one of the Lanterns any more than a figure picking up a Hammer from the Book has to be a Worthy figure... Resources are universalized where any figure can use it. Heck, the USS Enterprize (Star Trek Clix) can be assigned a Construct from the Battery, as could Hal Jordan, Sinestro, or Nekron (all 3 different Lantern colors).

I appreciate the clarifications.

So if I understand it correctly, I can build a non theme team with the orange battery, a orange lantern, orange ring and then add Mephisto , assign a construct with the lantern to Mephisto, and unleash the nastiness.

Diabolical

Thank you all!

It won't be a themed team, though.

When you construct that team, which is when theme is determined, Mephisto does not have the proper keyword. He will get it once the game has begun and you assign him a Construct from the Battery, but by then it is too late to make a themed team.

Just wanted to to make sure you weren't expecting to get a bonus to map roll and themed team PC.

I always forget this part, that the character assigned a construct from the battery has that battery's assigned Ring's (Corps) keyword. This comes into play when Im using Lysa Drak and I forget that my Green Lantern Kyle Rayner's construct from my PB has a BL ring in it and gives him a shared keyword with Drak. Theres been many times I forgot about that and coulda used her prob control to good effect when Rayner was attacked.