I see 'race' and 'classes' as the same. It's good to focus on one, and with banners get a second one.
With experience you'll find that some bonuses are good enough if you have 2 of them (mage, fae, beast, elemental, horror), 3 of them (demon, dweller and bug) and some you need all before it becomes really good (skeleton, healer, brawler).

I recommend not going for the difficult ones like Marksman, Ghoul, Stalker and Vanguard, since they're very situational and require some experience before you understand when and where to employ them.

Hope that helps :)

I just started playing this via Humble.

From my experience so far, I get farther when I try to make the most out of the best thing available rather than try to force a particular build idea. You can burn through a lot of good rolls hoping to get one more X-type unit.

There is no army composition that will always win.

So all options you mention are equally viable or indeed equally flawed.

Invest your resources smartly. You can compensate your weaknesses and augment your strengths a great deal with clever item placement.

Well, actually if you play your cards right and focus heavily on banners and perhaps get the Skull of Hedra, it's possible to get combinations of three maxed out banners and sometimes even four. For instance, you could get Skeleton, Mage, Marksman. Of course it's very powerful if you can get all three, but it's not easy.

Duplicators exist. This is why. My last run (2nd ever) was built almost exclusively on duplicators and a few shop finds. Spend the coins on the vault, not rolls.

Instead focus on Tank/Fighter/Caster first.

Build synergy (race/class/etc does not matter)- Simple recipe:
Get something to absorb damage
Get something to deal damage (arguably most important due to escalation)
Get something to heal (sustain)
^Tie them all together with traits/items/events etc (synergy)

Elementals for example are great at covering all of these and do so across the rarity values (important for many reasons). Skeletons on the other hand are very "common heavy".

You can definitely play your own strategy - the power rolls duplicate what you already have and there are tons of ways to get what you want/need without praying for rng.

Set your intention/strategy early.
Then adapt as you go.

More important that your units is how you equip them...

Monty for example isn't that good (common), but he is a defiler with unlimited range and a focus on attack speed. Therefore I like to give him FOECLEAVER and tons of attack speed.
^You don't need full reptile or full defiler -- that thinking is a noob trap. Instead you can throw in Fester (defiler/ghoul) and slap a ghoul banner on Shelly (reptile tank) -- just one of many viable combinations (doesn't matter).

Craggus is an uncommon - and a vanguard/elemental that you may overlook due to trait combos. He's almost always a solid choice because his skill scales with HP and Armor -- which is exactly what you want on a tank. Getting him more tanky (armor) even increases his skill damage directly.

Savage Crow is right, focus on Tank, then healer, then DPS, not so much synergy at first.

Try: Ancestry - Mountain Dwarf + Occultist - Forbidden Path.

I tend to aim to focus a single potent target. So for instance, the common bug duelist that jumps in is a wonderful hyper-carry target. Stack up bug to 5 point, put on high spell power and life steal, and he'll just jump through the targets, eating aggro with huge shields, lifesteal and damage resistance. And it's easy enough to invest in a "backup carry" of the bug/horror that with proper itemization/tile buffs, get his spell to 0 mana cost. Or get that one spell that you can slap on a unit from the hex city event that gives all of your guys +spellpower and Attack speed per cast. That one is fun with mushrooms too.

And River's buff choice that halves the cost of banners is pretty insane. They go from the price of a combination item to less than a component item. Considering how River is quite frankly the hardest of the final bosses IMO (I've lost more to her than any other by far.
4 maxed out class/racial buffs is just absolute hell if the RNG is remotely bad for you on the combo it gives her), ensuring she isn't the final boss while also getting those nuts banners? Hard to choose anything else.

The struggle (if any) is typically in the early game.
I highly recommend Zamfir due to that. He's def my favorite.

Zamfir provides:
Class Berserker = A great (random) starting item
Ancestry Satyr = Great early buff (works endgame too if you avoid variety)
Forbidden Book = Ban 5 you don't want
Forbidden Path = Huge buff until you build your synergy/team (gets you those early wins)

For a fun run in general I've been enjoying Scrappers.
Mikki can tank tourney as a T1 with proper gear (20k+ damage taken - no death)
Ian can also do 20K+ damage and is one of the best commons in the game.

For a fun (hard) run I recommend Kobold (pure commons - no mythics) -- really helps you appreciate the common class and find strengths/synergies without having access to full trait rosters.

Edit: My last kobold run was especially fun (crazy) as I had 8 slots (kobold+forthepeople) and 3 items that added extra units (11 units in play + 4 item slots *you can have 5*)

Just completed an Elemental/Healer run yesterday. War banners helped a great deal, but there're enough ways to get the missing minion to get full bonuses.
I too thought I had to complete those race/class traits to win but I've completed a few successful runs without any active traits (not even a trait requiring two minions). At least, in vanilla difficulty, I have to test the other challenges.

And by the look of what is available in perks&items, you should be able to solo a run too.