proper way to approach learning higher (university level) mathematics I have been struggling to self-learn some somewhat higher mathematics- mostly university level mathematics. However, I've looked up other questions and they didn't mostly line up with what I am personally struggling with.

The problem I am facing simply put is that most of the math books at this level rely on proofs and examples more than tedious repetitive- mostly computational- exercises at the end of each section/chapter like in earlier math subjects. Problem is, I feel frustrated that I can't verify my solutions OR WORSE reproduce the proofs I analyze and study. Sure I can understand the proofs just fine (most of the time), but revising them the next day feels harder let alone reproducing them without taking a peek. I feel hesitant on whether I should take the material as is and move on to the next section/chapter or fret over the material until it becomes a second nature (probably takes weeks and highly inefficient; one page per week at worst). 

How do you think I should approach this without feeling like I am skipping or just spending my time inefficiently? please advise.

Note: English isn't my first language so when I write proofs I tend to be redundant and less compliant to the common writing formats.
Advanced mathematics tends not to have "tedious repetitive - mostly computational- exercises at the end of each section/chapter". Rereading abstract proofs is probably not a good way to use your time. Spend as much as you can on the examples. Take each theorem and see what it says about all the examples you know to which it applies. See how the proof works in each particular case. Get several books and mine them for more examples. The ones that occur in all the books will be the important ones. The rarer examples will be instructive too.

Study counterexamples too. See how theorems fail when one or more of the hypotheses is false.

You need not master section $n$ before reading ahead into section $n+1$. You can and should circle back from time to time. The earlier material will probably become clearer that way.

(Your English here is just fine.)
University level...I think it depends on your current nation and your field of study as well.

For example, I learned in my engineering department in the university in Republic of Korea these subjects;

ODE: ordinary differential equation

PDE: partial differential equation

One of books widely used in the engineering course is Advanced Engineering Mathematics written by Erwin Kreyszig. This book could show you what subjects are taught in the university around the world.