Apart from having enough food and water, a strong defense (higher than food + water total to minimize attacks) and a general absence of attacks on your community, the happiness is mainly controlled by decorations and luxury.
Adding televisions, rugs and non-essential furniture (anything other than beds) to your settlement will increase the happiness of people living there. Basically, if you try to furnish their homes with care, they appreciate it. Who would have thought?
In one of those endless "How to foster your settlements" guides I read that apparently, the most efficient way to raise happiness in terms of resource/performance is packing any and every vertical surface with paintings. They require only wood to build which should be in overabundance and don't take up space on the ground for essential things.
Here are the different ways I've been able to find fusion cores so far:
Merchants - At least a couple merchants I've come across sell fusion cores. This currently appears to be the easiest way of obtaining cores if you have the caps.
Fusion Core Reactors - Not sure if this is the right name but they are big light green generators (like the one you use to get a fusion core in the mission where you meet Preston Garvey) and are found in many buildings throughout the Commonwealth and always give 1 fusion core. The difficulty in obtaining cores in this way varies but if you're an avid explorer you who searches every building (like me) you should have a good stockpile of cores in no time.
Power Armor - At least a couple of Power Armor suits I've come across have already had at least a partially charged core. Since you only need to power one suit at a time I make sure to take the core out of suits I'm not using. Also, if you equip a companion with power armor they will need a fusion core but don't seem to drain it so it's best to just give them one with low charge.
Sentry Bots - These guys are pretty rare and super hard to take down (the one I encountered was hit with 2 mini nukes and still had to be finished off with a laser rifle), but they drop 2 fusion cores and some other goodies. Theoretically, if they respawn somehow you could farm them for cores (albeit at a cost of ammo and stimpaks).
Random Places - I have been able to find a few cores in random containers or just lying on a table or the ground but it's very rare. High tech or military related buildings and locations seem to have a better chance but I've also found a few in people's houses. Not sure if the chance of finding cores is increased with luck or with some perk but it's definitely worth investigating.
As far as the most efficient way to get cores, I think that depends on your play style. Just exploring as many areas as I can has worked out pretty well for me; but if you're more charisma oriented then getting a deal on cores from merchants might be best, or if you're focusing on combat then tracking down sentry bots or other rare, hard-to-kill baddies could net you a good amount of cores.
As a bonus, here are also a few things you can do to extend the life of a core:
Repair Bobblehead - Found at the top of the Corvega Assembly Plant, it permanently increases the duration of your fusion cores by 10%
Nuclear Physicist Perk - With each rank it increases the duration of your fusion cores. At full rank it doubles the lifespan of your cores.
Fusion Core Reactors??? - I've heard a rumor that placing a core back into a reactor and pressing the button next to it will allow you to recharge the core. I haven't tested this myself but if true it could help alleviate the issue of finding cores in the first place.
Best Answer
Short story
Pocketed is better for carrying capacity. Lighter Build helps with stealth and AP, but not nearly as much with carrying capacity.
Long Story
The Nukapedia page Fallout 4 Armor Mods has weights and effects listed for various pieces of armor. As of writing this, it was not completely filled out for all items. It does have all the information for metal chest armor, so let's look at that as an example.
Polished Metal Chest Armor weighs 13 lbs.
Lighter Build reduces the weight by 3.2 pounds.
Pocketed increases the weight by 0.8 pounds, but adds 10 carry capacity, for a net gain of 9.2 pounds.
Ultra-Light Build reduces the weight by 6.4 pounds.
Deep Pocketed increases the weight by 0.8 pounds, but adds 20 carry capacity, for a net gain of 19.2 pounds.
So, clearly, pocketed is much better for carry capacity in this case.
Scanning through the other armor, the exact difference between the improvement provided by the pocketed mods and the improvement provided by the lighter build mods varies, but in all cases it appears that the pocketed mods are significantly better for carry weight. This applies to both chest pieces and limb pieces.
Perk Efficiency
You asked whether it would be better to get a higher rank in the Armorer perk or just put more points in strength. That's a pretty complex question, that will depend a lot on subjective preferences. This is because Armorer unlocks a lot more than just the Deep-Pocketed mods, and Strength does more than just increase your carrying capacity.
That being said, if you just want to maximize carrying capacity, you should invest in Armorer.
It looks like Deep-Pocketed only requires Armorer 2 on most, if not all, types of armor. Two points of strength will only gain you 20 lbs of carrying capacity. Putting those two points into Armorer instead allows you to put Deep-Pocketed on all your equipment, gaining you more than 20 lbs on any set of armor.