Most anti-biotic syrups do have sugar in them, as the antibiotics themselves taste pretty foul (with usually a strong smell)! Sugar-free options are available (but not for all types). Some types of antibiotics may cause a bit of hyperactivity (although very rare). Bactrim is one of them (especially if out in the sun).
5ml 3 times a day is a pretty standard dose and depends on the weight of your child. Antibiotics need to be taken at regular intervals (in this case every 8hrs) for them to be effective - so I would not reduce the dose to try and reduce the after-effect.
It could also be the infection itself that is causing the behaviour with DD. High temps and infections can have this effect on kids. Don't worry too much unless it is really affecting the health of DD.