Tl;dr: Bringing Intelligence up to 20 is necessary at some point, but doesn't have to be immediately. War Caster and Resilient are both great options and will help you maintain concentration. Heavy Armour Master and Lucky are also great options but won't have much impact on concentration. None of the other feats seem particularly worthwhile for you. Personally, I'd probably take War Caster and then decide between Heavy Armour Master and bringing my Int up to 20.
I'll address your points one by one, starting with:
Bringing Intelligence up to 20
You definitely want to bring your Int to max at some point, because it controls your primary means of attack. The question is whether you want to get it now, or leave it till later. With that in mind, consider what 2 more points of Int actually provide:
- +1 to hit with your spell attack
- +1 to DCs for your spell saves
- +1 to Int saves
- +1 to Int checks
- +1 to number of spells you can prepare
None of these are huge benefits in the short-term. Long-term, this makes you 5% more effective at most of the things you specialise in. (Warning: gross over-generalisation.)
War Caster
You don't intend to have both hands full, so this only gives you advantage on Con saving throws to maintain concentration, and spells for opportunity attacks. However, both of these are strong benefits. Advantage to maintain concentration needs no explanation, but spell opportunity attacks might not be as clear.
To start with, you're planning to be in the thick of the melee, so you should be getting opportunity attacks often enough to want to use them. Unfortunately, you aren't planning on using a weapon, and your signature spell Vampiric Touch won't work with opportunity attacks. (It specifically requires an action on your turn to make an attack with it.) So in your present state, you have no use for opportunity attacks. Giving yourself an option could pay off.
On the other hand, opportunity attacks aren't the only reaction, and you might want to save your reaction for spells like Shield or Counterspell. That said, these require resources, whereas with War Caster you can always use your opportunity attack to cast a cantrip.
Heavy Armour Master
As you say, the Strength is negligible where you're concerned, so the question is the damage reduction. This is a great option for someone planning to be in melee - 3 damage off every non-magical weapon attack adds up to be a lot of damage very quickly.
For concentration, it's a bit weird. It won't affect the vast majority of Con saving throws you make to maintain concentration, since the damage has to be at least 22 for it to have an impact. (Many high-damage attacks are magical, and it won't affect those either.) When it does affect the DC, it will only reduce it by 1-2. So as an option to help you maintain concentration it's a bit of a bust. It's still a great defensive option though.
Keen Mind
I can't say any of these features have ever come up in a game I've played. They're all highly situational and I can't find anything to redeem this feat other than "it's kind of cool, I guess".
Linguist
3 languages is pretty good, but there are spells that take care of this sort of thing for you. I've never seen a situation in game where a cipher would've been handy. If you're looking for a way to send messages to someone that no one can intercept, there are a variety of spells to achieve that.
Lucky
3 rerolls a day that you can choose to spend once you've seen the result? This is huge. You can spend them on an attacker's roll against you? That is just nuts. If you get critted, you can just wipe it away. Note that its benefits aren't nearly as potent on rolls that you make with advantage, but they're even more helpful on rolls you make with disadvantage. As far as concentration is concerned, you can fix a failed concentration roll 3 times a day, but you're likely to be making a lot of them, so something that isn't a scare resource and benefits all of them is probably worth getting.
Observant
This feat is frighteningly good. Just not for you. For a character who's maxing Wisdom and has proficiency in Perception, this makes them a portable radar station. (One of my players, the cleric, took Observant and has been making my life hell spotting every stealthed enemy and trap a mile away.) For you, it will make you almost as good as they were without it. If your party lacks a "sensor" character, it could be worth considering. Assuming you have one, I don't recommend this.
Everything in the above paragraph is based on Perception, without considering Investigation. That's because Perception is used far more often than Investigation, and passive Perception is used almost constantly whereas passive Investigation rarely sees the light of day. This can vary from campaign to campaign, obviously, but in general Perception is better.
Lip-reading is a cool little feature that I have seen used several times in intrigue situations, but I don't think it's worth spending a feat on.
Resilient
For maintaining concentration, this is pretty strong. More reliable than the advantage provided by War Caster, and scales better as you level. As well as that, though, there are a lot of effects that trigger Con saving throws. Most diseases, poisons, many spells, and a lot of monster effects. There's a saving throw for every stat, but Con, Dex, and Wis are still the most common, so this is definitely worth your consideration.
Ritual Caster
First and foremost, you have every benefit of Ritual Caster (as far as spells on the Wizard list are concerned.) Second, the benefits of this particular feat are highly dependent on what scrolls the DM feels like tossing your way. So the only quantifiable benefit of this spell for you is learning 2 1st-level spells from another class's spell list. Unless you can see something you really want (I can't), I can't recommend this.
Besides that, most rituals are either available to the Wizard or can be effectively duplicated by the Wizard. The only one I can see that you can't produce for yourself (barring a Wish, obviously), is Silence, which I can't quite see how you would use as a ritual. (It's more of a combat spell, and the 10-minute casting time is inconvenient in combat.)
I'll try to get the simpler issues done first - and the simplest one is Great Weapon Fighting. You should not take this. If you're using a spear, it is strictly inferior to the Duelling Fighting Style. See my answer here for the analysis. It's especially true for you because you want to tank, and for a quick and easy boost to your tankiness it's hard to go past a shield.
The next issue is how many levels of Paladin to take. This one gets a lot more complicated, but I'll start with the basics. When multiclassing in 5e, you want to take either 1 level or a number of levels that is a multiple of 4. This way, you don't lose any of the vitally important ability score increases.
The next thing to consider is what you actually want to get out of your Paladin levels, and what each number has to offer you. You also need to consider what Bard features you're missing out on by taking Paladin levels, but you've already got almost all the Bard features - from here on out your existing abilities mostly just get more powerful, rather than you gaining new things.
Paladin level 1: Gives basically nothing. You get the multiclassing proficiencies, Divine Sense, and Lay on Hands. Not a useful number.
Paladin level 4: Fighting Style, Divine Smite, Divine Health, and Channel Divinity. (And 1st-level Paladin spells, for what that's worth.) This is pretty good - a lot of the Paladin's core features for a minimal investment. If you're taking 4 levels, I'd recommend the Oath of Devotion, since (I think) it has the best Channel Divinity options.
Paladin level 8: All of the above, plus Extra Attack, Aura of Protection, and either Aura of Devotion or Aura of Warding. (Technically you could take Relentless Avenger, but it's not that great, especially for what you're trying to do.) Also 2nd-level Paladin spells, which is cool because one of them can be Find Steed, which is basically a class feature masquerading as a spell. This is a good place to finish taking Paladin levels because the next really good feature is at level 15. If you're going this far, the Oath of the Ancients starts looking as good as the Oath of Devotion, because Aura of Warding is so powerful.
I'd definitely recommend either 4 or 8 levels of Paladin if you're going to do this - 1 just seems like a waste of time. As for what order to take levels in, I'd actually suggest waiting till you've got your next ability score increase to do this, because doing it without War Caster might be a little awkward. Obviously, if you want to get better in melee sooner rather than later, you should start taking Paladin levels sooner rather than later, and you can work around the problems with casting your Bard spells with your hands full, it's just tricky.
Best Answer
Looking here, and assuming all books allowed, though you're not interested in looking at the new Beserker, I recommend.
For a dwarf:
1: Dwarven Weapon Proficiency
2: Bludgeon Expertise (You'll want this for the avalanche maul) or Enduring Mountain (tribal feats are excellent if everyone buys in)
4: Improved Defenses (your ref and will will be low. This helps a little) or Bludgeon Expertise
6: Cunning Stalker (if you're off and about charging random people away from your party) or Mark of Storm (While it doesn't support an avalanche hammer, it's doubtful that your DM will give one to you. A lightning weapon is a lot more common, and a slide 2 on hit is excellent for making enemies stick to your defender and setting up your next charge.)
8: Deadly draw (if mark of storm) or Blood Cousin's Tribe (if people've gone enduring mountain) or Reckless Charge
For a Goliath
1: Goliath Greatweapon Prowess then, as above. Goliaths don't need superior weapons with greatweapon prowess.
For a Half-Orc
1: Bludgeon Expertise or Heavy Blade Expertise (depending on your style and whether or not you can request a badge of the beserker)
2: Weapon Focus or Enduring Mountain
4: Toughness or Improved Defenses
6: Cunning stalker or as above
8: Thirst for Battle
For a Minotaur
As half-orc, but 8 should be bloodied ferocity
For a Warforged
As half-orc, but 6 should be warforged tactics, and 8 reckless charge or go the mark of storm route
To generalize
Taken from the charop guide:
For a strcon barb, especially one with a non-trivial investment into con, you'll be really hurting. Your first feat, however, should almost always be iconic or accuracy-related. +1 and a bennie is the best way to inflict the best status condition there is: dead. However, before accuracy is the signature weapon of the barbarian. As a charger, you functionally have 2 enchantments to choose from in the heroic tier: Vanguard (+1d8 damage while charging) works on any melee, but a fullblade is your best bet for the increased accuracy. Whiffing a howling strike sucks. Or Avalanche Hammer Mordenkrad, for a lovely +1[W] on charging. With an avalanche mordenkrad, your charger will be doing 5d6 (+1d6 because horned helm) brutal 1... at will. It's therefore important to choose between weapon types as early as possible.
If you were a strdex barb (very much recommended, You don't need a secondary con to take advantage of rageblood) I would instead recommend a vanguard gouge, so you can take advantage of a fighter multiclass for surprsing charge and spear expertise. +scaling to damage from the expertise,+1[W] with surprising charge with combat advantage (gained from cunning stalker) makes you ... remarkably scary.
If you want your DM to throw dice at you, I recommend mounted combat and a dire boar. Dice will be thrown, however.
Enduring Mountain is something I've got experience with in my own games. It grants a remarkable amount of health per surge used if everyone buys in and helps make up for a lackluster leader. Or for the depressing healing demands an almost naked strcon barb imposes.
Improved defenses is generic, but useful. When you have slots open up, it's worth retraining into superior X (or just grabbing superior will on top of it.) Grabbing superior reflexes is also nice on the combat advantage charging chassis for that excellent first-round combat advantage.