Wall of Text Incoming
For the DLC missions, I found each one had it's own "point," so to speak. Lonely Hearts took place in what I considered to be a very beautiful environment and I enjoyed the change of scenery. I enjoyed Dead Money for it's seemingly "survival horror" angle. It was probably the only mission the second time I played this game that actually gave me trouble. Old World Blues was definitely the most fun in my opinion and had me laughing more than any of the others. Finally, I found that Lonesome World had the best story in that it somewhat helps to create a picture of the Courier pre-Benny.
If I were to play through them again, my order would go from the one I least enjoyed to the one I most enjoyed, so while I enjoyed all of them, it'd probably go Lonely Hearts, Dead Money, Old World Blues and then Lonesome Road. I technically enjoyed Old World Blues more than Lonesome Road, but in my opinion Lonesome Road rounds out the Courier's story nicely and so I'll forego my usual rule to put that one at the end. Having said that, this is just my ordering due to enjoyment, it probably doesn't make much sense story wise.
As for ED-E, I agree with Eclectic in that you should choose based on which faction you want to help more and what you want ED-E to look like. First time playing, I went with the Brotherhood because I liked the look of shiny ED-E, as they remove the license plates from the chassis. Second time through I went the Followers because I prefer the Followers and their willingness to share what they have. Ultimately, both of my Couriers could tank enough damage and deal enough that ED-E's upgrades didn't make a lot of difference either way.
For a melee/explosive/unarmed character, I can sort of help. The Courier I'm playing now is melee and unarmed, but not explosive. Having said that, once I got the katana and super sledge the unarmed half sort of dropped away. As far as S.P.E.C.I.A.L goes, I'd advise a high strength, endurance and intelligence. Strength and endurance should be obvious, so you can get to the enemy before they kill you and so you can do a lot of damage when you get to them. Intelligence I always put to 10 on my characters as it lets me maximise the amount of skill points I get. By level 50 my courier has 100 in everything, except barter, energy weapons, explosives and survival, which are 49, 92, 23 and 81 respectively. Granted, there are certain skills that have been increased with perks and whatnot, but that should at least give a rough idea. These I put really high at the cost of perception and agility, as a melee character isn't using much V.A.T.S and ED-E's enhanced sensors perk lets me see all of the enemies on the map anyway. Of course, you might want to take a hit to charisma or luck instead of perception, as perception is the governing attribute for explosives.
As for perks, I always suggest Jury Rigging for every build because it is a great way to earn money and to keep your really powerful and probably very rare weapons in good condition. For the melee I'd also highly recommend virtually all of what Shanicus recommended, in particular the Adamantium Skeleton, Toughness, Piercing Strike, Unstoppable Force and Super Slam perks. Especially Super Slam. With the crazy high attack speed of the katana, you can almost be guaranteed that any enemy you come across will be knocked down. I can take 3 Deathclaws in the Thorn thanks to that and Piercing Strike alone.
Hope that's helpful in at least building the melee side of your melee/unarmed/explosives character. As for a name, go with Sir Billingsworth, because it's the first name that popped into my head.
TL;DR
1) Play Dead Money next. 2) ED-E's upgrade is nearly unnoticeable in the long run so go with your favourite faction. 3) Go high strength, endurance and intelligence. 4) Get Jury Rigging, Piercing Strike and Super Slam for your perks. Keep in mind these suggestions leave out the explosives side. 5) Name the character Sir Billingsworth.
Edit: Fixed a typo