Removed wainscoting - Will I need to sand the entire wall?
It looks like a sanded finish, mix a little paint with some fine sand and dab it on the smooth spots with your fingertip, it won't be a perfect match since the original has a number of coats of paint on it but it will break up the smooth. No need to sand the whole wall, Wipe the spots you touched up with spackle with a damp sponge to remove the excess spackle leaving only the smooth area before you apply the sand/paint mix.

Don't concern yourself with textured rollers or anything like that. What isherwood says about the rollers will work very well. You will need about a 1/2 cup of you wall paint and maybe a tablespoon or two of fine sand, that much is ON the floor of the big box stores where the play sand is kept, make sure it is like table salt, not rock salt or even coarse salt and mix it in the 1/2 cup of paint. Like I said, it will not be a match for the texture completely, but it will help blend in the patch to the surrounding area. Again, hard to reproduce what years of paint will do to a surface. 

I need to stress that this is only for the areas where the small blemishes are, not for general painting of an area. Painting of texture over texture will give the results you would expect. If and areas have been sanded, which I think the band that is painted white looks as though it was sanded or scraped pretty hard, this may show up differently. If an area that big is painted with the sand mixture, it will look worse than what it does now, if I see it right. If it has not been sanded or scraped, great, just touch up the small blemishes you already filled. (I am thinking nail holes and such.

Again to stress the treatment of the blemishes. Spackle always covers a larger area than needed. It is just the way it goes. After I do a wall repair on a textured wall, I wipe/wash the repair so the spackle (I use joint compound) so all the excess is gone from the surface of the texture, well maybe a little stays... and that will leave only the immediate area of the repair to re-texture. Brushes or rollers are too big to do this type of touch up and will get the new texture on the original and make the repairs more detectable. "Touching it on" with you finger will or should give you complete control how big or how thick or how much you apply on the small blemishes.