What should I do about walls that are shiny and dull at same time?
A picture would help but...what you are describing sounds like flashing to me.  This is when the sheen of the paint is not consistent (it has nothing to do with the physical even-ness of the surface).  Flashing occurs when the paint absorbs into the surface unevenly, thus it is common when dealing with patching.

Step 1:  If the walls are really rough all the way around skim coat everything to get an even base to start with.

If the walls are only bad in spots simply mud/spackle and sand the spots

Step 2:  Prime anything that is not paint.  Use a good sealer primer, I would suggest doing all the walls judging by what you have described.  Let the primer cure (this is not the same as drying).  Cure time should be mentioned on the product label.  I would recommend you use an oil base primer.  Yes its smelly and tough to work with but it does a much better job of covering stains and drying evenly, it also penetrates better due to its longer dry time.  Sand lightly with a medium grit sanding sponge once the primer is cured.

Step 3:  Paint.  Use two coats of paint.  Cut the edges all the way around first, then come back and roll the large sections of the wall where your cut work has already dried.  The better the paint the better the result.  Good paint is expensive but totally worth it.  It is easier to work with, splatters less, and looks better when you are done.

Personal opinions that you can choose to ignore if you wish

Avoid anything that claims to be a primer/paint in one...they are all garbage.

Good paint and good brushes/rollers are worth the cost, especially since you can clean and reuse them. (I will note that using cheap stuff with oil based primer is perfectly normal and I do it myself to avoid the hassle.)