No card in the game is cash-exclusive, though I think there are some event-exclusive ones (which might not be exclusive at all) and dungeon-exclusive ones.
In fact, some cards are synthesis-exclusive. The cards from the most recent episode (in this case, it's now EP6 which was just released a few days ago) are not made available in the Shop, meaning you can only obtain them through the Lab.
Training, as far as I can tell with what I've been messing around with, consists of two phases.
The first phase is the EXP Training phase. Here, you can choose a card to train over time for EXP, gaining roughly 10 EXP per hour. There's an item you can apply here but I have no idea what it does. Maybe it speeds up EXP gains.
After a unit accumulates EXP, you can take it to the second phase, known as Enchanting. This is where your card gains stats. Enchanting a unit requires two things:
2. Extra copies of the unit
Usually it costs 100 EXP to enchant a unit, but I've heard that this number increases later on as the unit is repeatedly enchanted. Note that you DO lose the EXP, so if you gave your unit 240 EXP using the 24-hour training option in the EXP Training phase, you'll only have 140 EXP left after you Enchant.
Additionally, Enchanting requires extra copies of the unit (the extra copies don't have to have any EXP or Enchants). The number of extra copies needed starts at 1, and increases by 1 with each successful Enchant.
There's also an item you can apply here but I have no idea what it does. Maybe it increases Enchant success rates or lowers the amount of extra copies needed, IDK.
Oh, and, BTW, you CAN fail Enchants. Or, at least, that's what I think. Because I tried enchanting my Tailor Maid once, saw no increase in stats, but I still lost the extra copy and the EXP. (The no. of extra copies needed didn't change though, so I'm thinking it only increases on successful Enchants.) I've enchanted my 2-cents Maid twice though (once for +1 HP, once for +1 STR (no idea how to pinpoint stat gains, I think they're either random or follow a set pattern)), without any failure.
Also, totally got my Crimson Witch Maid Siniah three days ago.
Oh my god what a beast. Her ability is:
"When this unit attacks, subtract the number of Academy cards in your hand directly from the HP of the defending unit. Heal this unit by that same amount."
Basically, if you have a hand full of Academy cards, you heal +5 HP and add +5 damage with each attack. Once she gets rolling it's very easy for her to snowball to 30 HP. She doesn't even need extra attack buffs because that autopierce ability combined with her base 6 STR is strong enough to one-shot almost anything.
My "strategy" now just consists of using my hand shuffles or just playing random cards until I draw her. Then she solos everything.
Pic is my opponent losing right after Siniah completely steamrolled over the rest of the enemy's deck singlehandedly. I didn't even use buffs spells.
But right now I'm messing around with a Dress Up deck. It basically uses the Dress Up cards from every faction (so sorta like a multisphere file or whatever). What they do is, they're strong cards with strong stats, but have 10 Size (meaning if you field them, you can't play anything else at all). But you can special summon them in certain ways that gives them Size=5, and usually something like STR+3 HP+3. So they like, transform.
Then you can use support spells that are 1 Size that have instagib effects if a Dress Up card is on your field (Conditional use: a unit card with "Dress Up" in its name is on your side of the field. Send one enemy card on the field directly to the grave).
Pic is the 1 Size cost "Dress Up Rise" spell instagibbing opponents' core units. Posted on the imageboards over there (they have imageboards...!!)
From the posters comments:
"I like to listen to Madonna's 'Dress You Up' while I smash noobs." (rough translation)
Haha, it's so fun.