1) As Chapuz said, the broadest definition is simply that a TCG is a CCG where trading is allowed/beneficial. Of course, it has far-reaching implications beyond this, which is the point of your next several questions.
2) It depends. In the case of elements, trading would destroy the game's economy, other than mark cards. The reason is that you could generate infinite wealth by creating multiple accounts. You could immediately trade the starter deck to your main account for a pittance. (1 electrum, or a pillar, or something) You could also do the same with oracle nymphs and electrum.
Some people have proposed preventing this by instituting only 1:1 trading (i.e. 1 card for 1 card, of the same rarity) to prevent people from creating nymph farm accounts, for instance. That would work to a degree; while it would maintain the -rarity- of cards, it would not maintain the uniqueness of cards. For instance, consider two players, Ann and Bob. Ann has spun six nymphs from the oracle, all of which are golden nymphs. She can make a pretty cool time deck with them, but has no other nymphs for any other decks. Bob, on the other hand, had a more even distribution of nymphs, with one each of entropy, death, gravity, earth, life, and fire. Thus, their collections are both unique, and both can make decks the other cannot, even if they acquire all of the bazaar cards and rares. With 1:1 trading, however, Bob can make a six golden nymph deck because his friend Ann is willing to 'loan' him her six time nymphs by 'trading' them all 1:1 for his nymphs, and then trading them back when he's done with his War match, or TPvP, or whatever else he's using them for. It also prevents people from creating dozens of nymph alts and just swapping nymphs around when desired between them.
Trading only works when there is a -limited- quantity of the thing being traded. In the case of elements, there is no limit. In paper TCGs, there is a limit based on the number of cards the manufacturer prints. In some online games, like MTGO, for instance, while there is not the same physical quantity limit, because every card created was bought and paid for by someone at some point, there is no way to generate free cards/dollars/ingame currency at will, barring illegal hacks or theft or something like that.
The only form of trading I can think of that wouldn't completely break the economy would be a 'conversion' sort of trading. For instance, taking the above example, Ann has six golden nymphs but no other nymphs. She doesn't want or need that many. Suppose that there was a feature where you could trade in two of any card of a given rarity in exchange for one card of the same rarity back. Thus, Ann could trade in two of her golden nymphs and get one of the air nymph she's always wanted, for example. Or, if that was still felt to be too strong, perhaps it could be that Ann could trade in two of her time nymphs for one -random- nymph in return. (and I would recommend excluding the type(s) traded in, to prevent the crappiness of being returned one of the same ones and thus literally wasting one for nothing) This wouldn't be abusable by multiple accounts, and it wouldn't let you just swap stuff around 1:1 whenever you wanted to. It would let you get rid of excess rares/nymphs if you had extras. Some people might have, say, 14 fahrenheits but only one eternity, just because RNG is... RNG, and it would be a way to even it out slightly. It would also reduce some of the grinding, because you could trade in some of those extra rares for that last evasive shard you need to complete trainer edition, for instance.
3) Booster packs. Boosters could work if done right, but if done wrong, they would either be inferior to bazaar buying, or else would let you get all the rares/ultrarares too easily. If boosters only gave you bazaar cards, there would be very little incentive to ever buy boosters over singles in the bazaar. People are obviously going to buy into whichever course of action is cheaper; thus, either the bazaar or boosters will be almost totally unused. Other than to finish off a collection, nobody would ever buy from the bazaar. In the worst case, if boosters were priced wrongly, you could maybe even sell cards in a booster for a profit, and just repeat until you got everything/got bored.
Now assume you can get rares in boosters. This would cause one of two things to happen; either people will stop trying to get rares from the arena, and just collect electrum for boosters, or else people will keep playing in the arena and ignore boosters completely. The players will choose whatever is easier when there are multiple ways to do something.
Now assume you can get ultrarares in boosters. There isn't any other way to get these right now other than oracle/tourneys, and the oracle cannot be controlled, so people, especially people with no current use for electrum like me, are going to immediately dump all of their electrum into boosters, sell everything that isn't a nymph/mark, and repeat until no electrum is left. While I do like the idea of being able to use excess electrum for something like that, unless it were very highly priced or nymphs were extremely rare in boosters, I think it would still end up just resulting in 'everybody has everything'.
Some of my arguments against boosters disappear if the bazaar is eliminated. However, particularly with a huge number (12) of elements, it would be hard for newer players to make much progress to start off. Suppose Bob starts with the darkness starter, for example. After playing for a few days/hours/whatever, he can now afford to buy his first booster. He gets no darkness cards in said booster, and gets mad. While this isn't necessarily a problem, it can be off-putting to new players, and with twelve elements, it can be very difficult to get even a few of the good cards in a desired element early. Contrast to other games, where there might be, say, five colours, or four factions, or three tribes, or some much smaller number than twelve, and the odds of a newbie getting something he wants to keep increase a lot.
4) I think this would result in pretty much the same thing as trading. Nymphs would just be auctioned off really cheaply by alt farms, and everyone would have them.
5) Pretty much already answered above.
