This is the sort of problem that anyone attempting to learn mathematics on their own will encounter. Over time, I've developed a few strategies that have worked for me. With regard to understanding proofs, I first attempt to prove the theorem on my own without first consulting the text. When this is unsuccessful, I'll read over the author's argument briefly to try to find the key idea and I'll then try to reconstruct the proof again. When I can't understand the author's steps I'll consult another reference. If I can't find another reference I'll post a question here. I also ask questions here that are more conceptual in nature that are harder to find references to.
With regard to exercises, it is unfortunate for the self-learner that most (higher) mathematics text are incomplete; they might have tons of great exercises but no solutions, thereby making it impossible to get a "second opinion" on your work or help you when you get stuck. Whether this is laziness on the part of the author or based on philosophical conviction is irrelevant; the fact is that this situation makes things far more difficult for the motivated self-learner than they need to be. So, my advice in this regard is to seek out more complete texts that have answers to at least some of the problems. It can take some effort, but depending on what you are interested in, you can probably find at least a few texts in this category. Indeed, many of the solutions to more popular texts can be found online as you've found; unfortunately though, this can be rather hit-or miss.
Problem texts however are also very good for exercise practice. The Linear Algebra Problem Book by Halmos, for instance, is an example of a very enlightening text in this genre. He teaches you just enough theory, just-in-time, to allow you to solve the next problem. If you can't solve the problem, he has detailed solutions to each one.
With regard to general analysis, there are many sources of problems with solutions. Alaprantis and Burkinshaw's Problems in Real Analysis, Solutions to Lang's Undergraduate Analysis and one of my favorite resources is Erdman's Problem Text in Advanced Calculus available freely from the authors website.
For basic algebra, perhaps Blythe's problem book Algebra Through Practice would be beneficial. I can't say though I like this one as much as the other texts I've used, but it's OK, but maybe a little too basic to get you very far. Dummit and Foot has great exercises and many solutions can be found on-line in some form.
The approach to the exercises is the same as with the proofs. I first give a serious attempt and only consult the solution when I'm stuck. Also, if I'm trying to get through more material, I usually "time box" my activity and agree with myself to "give up" after a certain amount of time. I really don't like to do this but I've found that if I spend at least 15-30 minutes on a "moderately" difficult problem that I still can't solve I can appreciate the solution a lot more when it is revealed.