Well I was able to beat Dheginsia on turn 1, on normal mode and without the use of any laguz royals after some thinking. Sadly I currently have no way of recording it. However, to compensate for this, I will tell you several tips to help kill him in turn one and I'll post the exact method I used. I thought up of several things that could help you kill the big D in turn 1. Mass blessed bolting, if you don't have enough boltings you could have your units move up to others who have it and already attacked then trade and attack again. Also blessed wyrmslayers helps you get Dheginsia into an early grave. Also, and I think this is the most important, the staff Rescue. The Rescue staff teleports an ally unit above the user. The most important thing about this is that if the person that was teleported hadn't moved then he will be able to move after the teleportation. This is very helpful in sending a blitz team to Dheginsia quickly however Rescue only has 3 uses so you better make your units hard hitters. Now I'll explain the exact method I used to down the big Dragon.
(As a side note I got this from Serenes Forest so I had to edit enemy locations because it was different in the Japanese version.)
First of all let me tell you the items and skills everyone had. (I'll only list the ones that are directly necessary for this method as well as locked weapons.) Ike had a blessed ragnell, a blessed wyrmslayer, and the skills Parity and Celerity (he also got the movement increase from boots). Shinon had a blessed Double Bow and the skill DragonFoe. Elincia had a blessed Amiti and the Rescue Staff. Pelleas had a blessed forged thunder tome (this was mainly to improve the hit rate to 105 from Thoron's horrible 65, also I got lucky and pulled the axe card so the mt of this tome was 1 point higher than Thoron's.) and the skill Nihil. Illyana had the blessed weapon Rexbolt and the skill Nihil. Mia had no weapon (I wish I had another wyrmslayer but I didn't so I had to improvise) and the skill Nihil. Micaiah, Mist, Boyd and Nephenee had nothing. These were the only units I brought for this chapter.
Ok now for the method. I had Boyd shove Shinon one space up and had Shinon move to square 1 and snipe the enemy below square 5 (yes I know it looks like a 6...). Now, thanks to the boots and celerity, Ike had a movement of 11 so I had him move to square 2 and attack him with the wyrmslayer while using the skill Parity. (Parity basically acts like Nihil except it cancels alot of other stuff too, I did this so that Ike didn't get raped by Ire.) Ike double attacked and did a total of 40 damage while he took 43 damage. I then had Elincia move to square S and teleported Pelleas to square T. I then used the remaining movement to move Elincia to square S1. I then had Pelleas move to square 3 (As another side note squares 3, 4 and 5 are all +10 defense squares. I know that 5 looks like a 6, blame paint. >_>) I then had him trade to get Ike's wyrmslayer and attacked Dheginsia with the thunder tome doing 23 damage while Pelleas took 41. I then moved Mist to square S, traded for Elincia's Rescue staff and teleported Mia to square T and had Mist use her remaining movement to move to square S2. I then moved Mia to square 4, traded to get the wyrmslayer and double attacked Dheginsia doing a total of 30 damage while she taking 39. (Mia had a C support with Pelleas, but I'm not sure if that boosted her damage or not, too lazy to check. >_>) Dheginsia now has 7 hp with one last unit left to kill him with...Illyana. I had Nephenee shove Micaiah one square up, had Micaiah move to square S and... er well you know the rest. After being teleported I had Illyana move to square 5 (stupid paint) and had her attack with Rexbolt doing 27 damage which, finally, brought down the big Dragon.
Gah that was a pain to type. Anyways I'm wondering if it's possible to kill Sephiran in turn 1.... Those stupid spirits of his keep protecting him everytime he's about to die. Eh... I'll think about it later.