My question regards the timing of the Team Hydra Ability...

If I have a Team Hydra Member at some point on the map...
Let's say I use a character's Running Shot or Hypersonic Speed (moving however many squares) and up adjacent to the Team Hydra Member...
If both characters have a clear line of fire to the target of my Running Shot, does the character performing the attack automatically get the +1 Attack Value?

Also, another scenario...
If I am carrying a Team Hydra Member and want to use Running Shot, after I end my movement but before the attack roll (now having a clear line of fire and dropping the Team Hydra Member in a square where it also has clear line of fire to the target) will I be able to get the +1 attack modifier?

Any and all help is appreciated.

Absolutely. The Line of Fire is checked the moment of attack

Again, yes. In the use of running shot, you complete the move before the attack is done. The same does NOT apply to transporter or hypersonic attacks

Why can't the same apply to Hypersonic or Transport Attacks?

Because in Hypersonic/Transporter attacks the attack occurs during movement, while the carried figure is placed back on the board at the end of the movement - it's off the board while being carried.

Ok, so if I use Hypersonic/Transport Attack, and before the attack roll I end up next to a Team Hydra Member (and both of my figures have clear line of fire to the target) then the modifier would be applied, same as the Running Shot scenario, correct?

Yes, it would. Same with Enhancement.

Except no LoS to the target needed for enhancement obviously.

HSS or transporter attack is a movement attack, so does not allow the character being carried (transporters cannot carry and use the transporter attack) will be in midair not on the ground so they are not adjacent, until the attack and movement is over. With running shot, it is move, stop or set down character, and attack.

With the plus with damage (enhancement, or Shield) there is no LoS needed.
With the plus with to hit (Hydra, or Police TA in DC) there needs to be LoS, but the character's range does not need to reach the target to give the bonus. Remember, range bonus damage-no need for LoS. Range to hit bonus a LoS is needed.
Also, know that enhancement can be used more then once. So multiple characters can use the bonus of enhancement as long as they are adjacent to the character in question.

And one thing to note that hasn't been mentioned yet is that the SHIELD TA can NOT be used if they are carried in a RS because the SHIELD TA requires that the figure take an action to activate.... NAAT (no action after taxi).