This has been a long time coming, but I think it was worth the wait. I wanted to cover all bases (pun definitely intended) before I answered these. Oh, and I think you're really going to like the answer to 2, which I have saved until the end.
1. This has now been amended, resulting in Unarmed at 1 and Melee Weapons at 2. As you quite correctly pointed out, we would be able to use our fists long before wielding a weapon, and so this order makes a lot more sense. It also matches up with the respective icons.
2. See the section at the end of this post!!
3. The previous suggestion of Ninja has now been replaced with Martial Artist, as described in the updated OP. This would fit far better than the overly stealthy Ninja Perk. Also, Including the Martial Artist perk in this category would definitely make sense. This perk would allow us to target specific body parts when using Unarmed or Melee weapons in V.A.T.S., and so this would have an inevitable impact on damage output, including bonus damage for blows to the head. Including this under Strength would also keep all things related to Unarmed and Melee in one place, which ties in with the fact that every point we add to the Strength stat would further increase damage when using either of these skills.
Also, the perk at 10 Agility, which I had at the time incorrectly identified as V.A.N.S. (which is now placed at 5 Endurance as described http://www.gamesas.com/topic/1529102-skills-perks-revised-edition/page-4#entry24211796), has now been replaced with the far more stealth-sounding Ninja Master perk. This takes into account all of the previously noted stealth abilities, as well as the icon itself, which shows VB running (which explains the connection to Silent Running) with his hand raised in front of him. Due to the image being very unclear at this stage, his hand could also be seen as being a Shuriken star.
4. Science! has now been moved as per your suggestion. This means that all skill perks now have a requirement of 1-6 in their respective stats, depending on the power and usefulness of the skill. Here is my explanation for the new lineup as posted earlier on the thread:
"I have moved Science! to match it's corrected requirement of 6 Intelligence. This would now place Science! and Acrobatics as the joint most powerful skills, as they would both require a 6 in their respective stats. This makes perfect sense when you consider what these skills actually unlock (note that Plasma weapons appear to have both ballistic and energy damage). The next most powerful skills would be Explosives and Local Leader, both requiring 5. We then have Hacker at 4, Gun Nut and Sneak at 3, and all remaining skills at either 1 or 2. Note however, that this does NOT suggest that either Unarmed or Melee Weapons would be in any way underpowered, only that they would be accessible by almost anyone from the start of the game."
5. I have removed both Chemist and Fast Metabolism for the reasons outlined in your post.
6. The Attack Dog perk is due to be fully revised in line with new information posted throughout this thread, staring with http://www.gamesas.com/topic/1529102-skills-perks-revised-edition/?p=24208488 post.
7. Damage from ranged weapons is one point that has caused a lot of debate on this and other threads. My first feeling was that weapon damage would be tied directly to the Gun Nut and Science! skills, but as many have pointed out, this would not make any sense with the skills being listed under Intelligence. A far more sensible option, suggested by several people in this thread (and which I will add into the OP), is that weapon damage would more likely be tied to the Gunslinger and Commando perks.
Even though these perks are listed under Agility, they would be the perfect place to add incremental weapon damage as well as ranged accuracy. This would affect all pistols when using Gunslinger, and all rifles when using Commando. And so even if you did not meet the requirements to modify for example, a plasma rifle, with all 4 ranks in Commando you would still be fairly accurate (both in realtime and in V.A.T.S.) and would be able to do a great deal of damage. If you later unlocked all ranks in Science!, you would be able to turn the basic plasma rifle into a far deadlier weapon. This would be further enhanced if you also took Sniper, which would progressively improve V.A.T.S. targeting when taking headshots, and would mean you could end up taking some very long distance shots with a sniper rifle. As a final point, the above scenario could be achieved with conventional rifles with an Intelligence score as low as 3.
And now, back to point 2:
In your post, you suggested that the Vault Boy in the 2 Perception perk is pitching to the one in the 2 Strength perk, as a direct reference to https://en.wikipedia.org/wiki/Fenway_Park, which we know to be the location of Diamond City. The problem I initially had with this theory is that the icons did not seem to match the suggestions of Little Leaguer and Heave Ho! Also, this second perk does not have any direct or indirect correlation to Perception. The final issue was that all existing perks carried over to Fallout 4 seem to have derived from Fallout 3, and not from other games. This would make sense, as Fallout 3 is the only other game in the series that was actually coded by the team at BGS, and so will have by far the highest number of references in Fallout 4.
As you know, the perks I have suggested for these 2 locations are Melee Weapons under Strength (corrected as detailed above), and Far Sighted under Perception. The reason I chose Far Sighted is due to the telescope which can be seen in the icon, and also the historical significance of both exploration and astronomy in Boston. But what I had missed at the time I first posted the description was just HOW significant that reference was. To explain more fully, I have quoted below the new descriptions for both Melee Weapons and Far Sighted as added into my OP a short while ago:
Melee Weapons (requires 2 Strength)
Allows you to use all hand held weapons, including baseball bats and many others, more effectively. Ranks 1-4 gradually increase the speed and damage of individual attacks. Note that increasing Ranks also unlock new mod types, including Barbed Wire, Razors, and Saw Blades. This perk comes with it's own bobblehead hidden in the game world that will unlock the next available rank when found.
Note: the icon shows Vault Boy about to hit a baseball, pitched by his neighbor, using the baseball bat raised up to his side. This would seem to be a reference to Fenway Park, the location now known as Diamond City. See the description of Far Sighted for more information. The direct connection between this and the neighboring icon was first identified by The Leprechaun in the thread linked at the top of this post.
Far Sighted (requires 2 Perception)
In this image, Vault Boy is seen looking through a telescope. This means that the perk will most likely allow us to briefly 'zoom in' on distant objects or locations in order to get a better lay of the land, and may also allow us to identify distant enemies. This would most likely require us to be standing still. Increased ranks would allow for a greater level of 'magnification' when using this ability.
Note that the use of a telescope in this image is most likely a reference to Astronomy at Boston University as seen http://www.bu.edu/astronomy/about-us/early-department-history. Note also that VB in this image appears to have pitched the ball to his Melee Weapons neighbor with the baseball bat. This would make sense given the close proximity of Boston University and Fenway Park, as seen https://www.google.co.uk/maps/place/Boston+University/@42.3504997,-71.1053991,14z/data=http://forums.bethsoft.com/topic/1529102-skills-perks-revised-edition/!4m2!3m1!1s0x0:0x2b346e00e0a3bec8. This would also make Boston University a prominent in-game location. The direct connection between this and the neighboring icon was first identified by The Leprechaun in the thread linked at the top of this post.
So there you have it. And it was right in front of us the whole time! This reference could well have a massive impact on gameplay, as it suggests that there is some significant rivalry going on between Diamond City, and whatever is now at the nearby Boston University. I think it may be time to expand my theory a little... possibly starting http://www.goterriers.com/trads/bost-trads.html and then perhaps http://i.imgur.com/N6yqGfI.jpg. I have linked this last reference to a brand new thread
With regards to this last point, please direct all feedback to the thread linked below:
http://www.gamesas.com/topic/1529468-super-mutant-terriers