Ranger would seem like the right choice, with Beast Master giving you a bear (for Misha), as well as some of his other animal compatriots over the years. The pet will want a lot of ranger levels, though, as it needs that for its stats. Maybe a light dip into Barbarian for some wildheart rage option (plus the dude is not exactly an armor wearer), or rogue/thief for more more bonus actions (and consequently, more off-hand attacks). Barbarian would be the other solid choice, given his big size and refusal to wear shirts.
1) Rexxar would be strength, because he dual wields axes, plus he's half ogre. So light axes until 4, when you can pick up Dual Wielder as your feat. Really, I think if you focus on dex, you're going to wind up more with a Drizzt than a Rexxar. If you intend to go all ranger, all the way, you'll have two other feats to pick from. You can use one of those to get Alert. Alternately, you can use gear to deal with the str/dex related issues. There's a bow early with +1 to initiative, another later with higher, and some other gear that boosts it as well. You can also use the hand slot, with the 18 dex bracers available in act 1.5, and later on, you can respec to take advantage of the Gauntlets of Frost Giant Strength to really play into his half-ogre lineage. Or you can go with a relatively low strength right from the start, and pick up the Club of Hill Giant Strength in act 1 to off-hand. Boom, 19 str. But then you can't put the Undermountain king dagger there, which you might want if you want to play into the crit aspect of the half-orc. But that weapon shows up at the same time as the dex bracers, so you can respec at that point.
2) Familiar can be useful for scouting and stuff, as well as creating advantage, but it isn't the strongest in combat, overall. However, that's all the more reason to be able to want to cast it without eating up your spell slots. If you don't want to go Beast Master, but do want an animal companion out, go for this one. Fire/Cold resistance can be relatively easy to get (there's a bow that gives you both in a single item), plus potions of fire resistance and such, but fire damage is equally common. Poison tends not to hit as hard, and it is more the saving throws you want to make to get rid of the disadvantage, which it won't help with. Any of them other than Urban Tracker are probably a good fit, though. As for favored enemy, both of those choices seem solid. If you don't plan to tank dex, then I'd probably go Bounty Hunter (since you'll have dex to add to AC, and those points in dex will probably mean lower wisdom). However, you'll cast the spell as a bonus action, which will mean you get less opportunity to attack with the off-hand (dual wielding kinda sucks in 5e, a few bits of cheese aside). And if you do tank dex, then get that heavy armor. That said, Rexxar himself is definitely not a heavy armor guy.
3) There are definitely good weapons to combine. Knife of the Undermountain King is a very common choice to off-hand, as an example, and it boosts your crit rate chance. And, well, if you want to be like Rexxar, I don't think sword and board is that. And yeah, I'd probably go with the feat, so you can use a bigger weapon in the main hand. Rexxar uses dual axes, and being half-ogre, well, they definitely aren't both light axes
So don't know about you, but I'd want a bigger axe in the main hand. I do recommend turning off the auto-off-hand use, though. Sadly, the axe selection is rather limited in the game, but there is one that adds +7 damage to critical hits, if you want to play up that aspect, and another that gives an option to make another attack as a bonus action after you crit (which would work nicely if you do go "sword" and board).
Overall, though, seems like you're trying to min/max things a bit. Like for me, if I want to play Rexxar like, that would be my focus. I'd have a party member whose main job would be to cast Enlarge on my guy for every combat, for example. Obviously less powerful than Haste, but half-ogre. Because being Rexxar like is what would make the playthrough unique. If instead, I were to focus on the min-max, the playthrough would quickly devolve into being like every other playthrough.
