The notion of a (left, say) Bousfield localization of a model category doesn't seem to be invariant under Quillen equivalence. There are a lot of things that could go wrong. But I don't know any examples. So if anyone could help me out with even one of the below questions, I'd appreciate it.


Let $M$ be a model category and $\mathrm{Ho} M$ its homotopy category. 


1. At the most basic level, what is an example where there exists a localization of $\mathrm{Ho} M$ which isn't the homotopy category of a left Bousfield localization of $M$? This isn't usually how left Bousfield localizations are presented (although it seems the most invariant way I can think of), so, as a bonus: what is an example where there is a localization of $\mathrm{Ho} M$ where the inverted objects aren't of the form $\{S-\text{local objects}\}$ for some class $S$?
2. More specifically, what is an example of (1) where the implied acyclic cofibrations aren't generated by a small set (actually I'd be happy to know just this, but of course what I'm driving at is that they shouldn't admit factorizations)?
3. Alternatively, what is an example of (1) where the implied acyclic cofibrations are perhaps generated by a small set, but not one which admits the small object argument (and they don't have factorizations)?
4. Pressing forward, if $F: M \overset{\to}{\to} N: U$ is a Quillen equivalence and $M'$ is a left Bousfield localization of $M$, then if the model structure $N$' induced by $M'$ along $F$ exists, it is a left Bousfield localization of $N$ and $F: M' \overset{\to}{\to} N': U$ is a Quillen equivalence. What's an example where $N'$ fails to be a model structure? Again it's interesting to ask about the various ways this could go wrong.
5. In the other direction, it seems dicier to try to induce a left Bousfield localization along a right Quillen equivalence. I guess that dualizing the argument from (4) will show that if the induced model structure exists, it is a left Bousfield localization and the Quillen equivalence restricts to a Quillen equivalence between the localizations. But it must be not unusual that this induced model structure fails to exist, right?
6. What is an example of a Bousfield localization that can't be induced from a Bousfield localization along some particular Quillen equivalence? How about two different Bousfield localizations that induce the same Bousfield localization along a Quillen equivalence?
7. Are there any model categories where *all* Bousfield localizations are known to exist, not just those generated by a small set?



Bousfield Localization and Quillen Equivalence
For (1)-(2), look at work of Carles Casacuberta. He has lots of good examples. His paper with Chorny on the orthogonal subcategory problem has an example for your (2), on the last page. This paper of Casacuberta-Chorny goes into great depth about (3) as well, and it led to Chorny's work on class combinatorial model categories and localizations (when the model structure is generated by a class, and you have a generalized small object argument). This paper may also include an example for (1). If it doesn't, I can dig out an example Casacuberta showed me last summer and send you a sketch by email (it's unpublished and not mine, so I don't want to post it here).


Part of my thesis can answer (4). You could have a Quillen equivalence of monoidal model categories $M\_1$ and $M\_2$, and suppose both satisfy the commutative monoid axiom so that commutative monoids inherit transferred model structures. There was a condition in my thesis that if the free commutative monoid functor Sym(-) preserves $C$-local equivalences then a localization exists at the level of commutative monoids. Since this condition need not be preserved by a Quillen equivalence, you can have a left Bousfield localization of $M = CMon(M\_1)$ that does not transfer to a left Bousfield localization of $N = CMon(M\_2)$.


For (5), I don't have an example off the top of my head, but I agree with you that I would not expect this to come for free. This would be a strange example, because on the infinity category level there's no difference between left Quillen equivalence and right Quillen equivalence, and there I do expect localizations to be induced (assuming everything in sight is presentable and accessible). So maybe look at some very simple model structures with a lot going on with the cofibrations and fibrations.


For the first part of (6), I see two ways to interpret this. Every Bousfield localization is induced by the identity functor, so I assume you didn't mean that. For the other interpretation, if someone hands me a Quillen equivalence and a Bousfield localization of one of the two model categories I see no reason at all to expect it to be induced by a Bousfield localization on the other piece. For the other part of (6), what do you mean by "two different Bousfield localizations"? If they provide Quillen equivalent model structures most people would not say they are different.


(7) was answered in the comments, by Phillipe.


Another paper you might enjoy is Casacuberta-Neeman "Brown Representability does not come for free." It's about a localization failing to exist at the homotopy category level, when it was expected to exist. 


