Favourite problem books at university level
List of problem books that may fit the definition in the question:
(1) Polya/Szego (2 vols)
(2) Halmos: A Hilbert Space Problem Book
(3) Lightman et al: Problem Book in Relativity and Gravitation
(4) Padmanabhan: Cosmology and Astrophysics through problems
(5) Arnold's Problems
(6) Chung and Graham: Erdos on Graphs
(7) Miklos Schweitzer Contest (2 vols)
(8) Garrity et al: Algebraic Geometry, a problem solving approach
(9) Radulescou: Problems in Real Analysis [thanks to @symplectomorphic ]
(10) Makarov: Selected problems in Real Analysis [thanks to @LeGrandDODOM ]
(11) Knopp: Problem Book in the Theory of Functions (2 vols)
This became too long for a comment, but it is only my viewpoint on your second and fourth question. I did include a fun problem to compensate for the long read.
I've been brought up with toy exercises as well. Even when I was 18 years old I never thought about anything to deeply. An answer either came quickly or I labeled it as too difficult and dismissed it altogether. When I was 19 years old, I first thought about how to find the roots of a quadratic equation since I had forgotten the formula. After 2 minutes or so, and writing stuff down, I 'discovered' the formula myself. I still forget the formula every now and then, but it takes ten seconds to derive the formula again.
Even though solving an actual problem feels good, I was used to toy exercises for the most part of my life. The first 3 years at university I was interested in mathematics but not used to solving anything beyond the routine exercises. I needed inspiring people and smart friends to help me on the right road and seriously think about problems. But unfortunately, 21 years of not having that mentality leaves deep scars. I am not the mathematician I could have been.
When you ask what students want, you probably are going to get two very different versions. Beginning students that have only seen toy exercises are unlikely to demand serious problems and work hard on them. You cannot really blame them since they know no better. The other answer is very likely to be a hindsight answer. In hindsight, thinking harder on certain problems and doing more work would have helped you becoming a better version of you.
Recently I started doing a Phd (with a large teaching assignment), I'm certainly not the most gifted mathematician but I do have my clear moments. Sometimes a smart insight occurs, but only after struggling with something and making my fair share of mistakes. This process is new to me and at times very hard. Having had only toy exercises myself for a long period of time certainly makes dealing with it during my phd for the first time a lot harder. This should not be the case!
More often than not do we hear old wise men say that things used to better. And, when it comes to educating mathematics, I have to agree. I see a lot of students struggling with exercises that even I did consider toy exercises myself when I was (a not so great) student myself. A vast majority did considerably less thinking than I did up to that point, and well, that's absolutely shocking. I try to motivate students and present them interesting problems that should help your overal understanding of mathematics. I try to tell the history of certain problems and the difficulties that great mathematicians had tryin to solve them. I hope that in this way I can save some students from not thinking, and not thinking is something that they were thought very well.
PS: How many numbers of the form $101$, $10101$, $1010101, \dots$ are prime?