6/8 means 6 beats in a measure and an 8th note counts as one beat. You don't always have to play any type of chord with just a downstroke (including power chords). It depends however, on the speed of the song and the sound that is being made. The slower (more arpeggio) a chord is, the more there will be a difference in sound between up and down strokes, because you can more clearly hear the sound either going from high to low, or from low to high. For power chords with distortion on an electric, you will get a "chunkier" sound with all down strokes, but upstrokes can be substituted for many reasons.
What song is this? Hearing it will be more helpful because depending on the tempo, it might be irrelevant whether or not you have to strum up or down.
As far as that measure, those bars with the short "double bar" are sixteenth notes. 2 sixteenth notes makes 1 eight note, or one beat. Those rests are also sixteenth notes. So that last bit of chord section is all 16th notes.
Here's how to count it:
1 & 2
& 3
& 4 &
5 &
6 &
Play on the bold counts and don't forget the rests
