Should a tooth bridge fit against the gum?

Yes. One design of a bridge does have it against the gum to give it the natural appearance of emerging from the gum as a natural tooth would. It also makes it harder for food and debris to get stuck between bridge and gum.
Almost always yes. The only time the replaced missing tooth does not touch the gums is either following an extraction if the dentist did not wait long enough for the gums to heal and shrink or in the use of a sanitary teeth that are attached to the support teeth but the underside sits well off the gums to prevent bacteria and food from accumulating and it makes it much easier to clean.
USUALLY,YES. The bridge was fabricated to fill an empty space as perfect as possible imitating your natural teeth. Sometimes for specific reasons, the dentist decides to leave a space between your gum and bridge, but not often. You shouldn't worry if the gum is not hurting due to excessive pressure against your gum.