[Gymschu]

You don't always need a "special primer" to cover semi-gloss. If you sand it rather thoroughly, wipe the dust off, then you can paint right over it. If you're not really into sanding it, you can use Zinsser's 1-2-3 bonding primer, allow it to dry overnight, then apply your topcoats of eggshell.

Your spackle problem is quite common. It's called "flashing." Spackle soaks up the primer/paint leaving you with dull spots. Always apply a coat of primer, allow to dry, then apply a coat or two of your paint to those spots, then apply your topcoats. That usually takes care of flashing issues.

 

[Gymschu]

I just use the end of my nap roller to lightly touch the spackled areas so the texture is the same. You don't want to leave brush strokes in the middle of a wall that also may flash just a bit because of the difference in texture between a brush and a roller.