Do songs that get used as openings/endings get sped up as well as cut?
As far as I know, theme songs are picked or created based on the anime's themes (if the song is appropriate with the anime's story) and its length will only be cut shorter while the tempo will stay the same (with the exception of course if the production decided to have another version of the opening to include in their OST).

However, most anime openings have faster tempo than endings to have that kind of "excitement feel" or some kind of "momentum" as an anime starts. But it really depends on the theme of the anime. If the anime has a tragic theme or a sad love story theme, sometimes production decides to have an opening song that has lower tempo, or a sad song, but mostly, it will have a faster tempo than its ending.