Games to collect stuff without leveling/fusing, similar to PvZ:H?

I can't quite hit all of those requirements. Closest I have is Ascension: Deck Building Game (aka what used to be called Ascension Chronicle Of The Godslayer for the name of its base game)

Turn based, and at least I find it VERY fun. Nothing on the wants list. OTOH, there's no daily quests or online content. Now that you mention it, when I say "deck building game" (DBG), I mean (real/physical) card games like Dominion, this/Ascension, and Star Realms where when you buy the box, everything you need to play is consistent and there. There are definitely expansions for those games, but those are self contained to. Contrast that to Collectible card games like Magic The Gather where you buy 3 booster packs, the first one may contain a rare, but the other 2 may not.

Each game is self contained (like a game of Chess), so nothing persists from one game to another. You always start each game with the same 10 cards.

 

Elder Scrolls Legends might be the perfect option for you. Part natural evolution of the Hearthstone formula (more sophisticated mechanics, less random, and more dynamic/flexible deck construction*), part unique elements.

* Even for a non-deckbuilder like yourself, this offer the advantages of a greater assortment of decks to pull from the web, a healthier meta with greater variety in the decks you will face. Plus, it makes arena drafts more diverse and fun.

And even better, it features a single player Arena (in addition to regular Arena) that offers a few unique advantages, such as resumable games (you can leave the game at any time, and the match state is saved online) - which helps play during spotty connection scenarios where connection loss would mean actual loss against a human opponent - varied lane conditions (much like Hearthstone's brawls, changes to game rules), and a more relaxed, contemplative, casual play style. (As much as I enjoy facing off against other players online, sometimes the tension of multiplayer is not desireable.)

And there is already a database brimming with decks to pull and enjoy. Add to this a more generous reward scheme than in HS (and thus, more collecting of stuff!), plus a single player campaign with story and ample rewards, and it really seems to tick off all yer boxes. And add some.

Only available for iPad right now, but Dire Wolf and Bethesda are working on the iPhone version.

DISCLAIMER: a handful of your starter cards do level up ONCE. But this is an automated process that only serves to introduce the player to the game gradually, and to create an early sense of player progress. It has no effect on normal gameplay, since the vast majority cards in packs and cards you craft have no levels, and the few that do are already leveled. You never draft the starter cards in arena, and they are never used in versus play. These are purely a tutorial mechanic, and never used again. Thus, it bears no resemblance to the tiresome card fusion and leveling of collective f2p games.

You should also really peek a sneak at Magic the Gathering: Puzzle Quest. A brilliant fusion of a collectible card game and a match 3. You collect cards and form (very small) decks of creatures, spells and artifacts, but you use match 3 mechanics for the actual battles, and generate mana to play your creatures, items and powers through gem matches. Once on the board, the cards interact with the match 3 mechanics, influence the board as well as your hand of cards, change rules of interaction, and damage your opponent.

The actual match 3-ing - with all its card-based permutations - is probably the most sophisticated and fun gem matching mechanics I've played with (next to the original, brilliant, brilliant Puzzle Quest).

And the collectibles is strong with this one, offering some of the most engaging, rich collection-based sense of progress and incentive to play I've found outside of proper CCGs.