Flowering cherries are susceptible to many problems, including cherry virus diseases, canker, twig blight, root rot, brown rot, powdery mildew, bacterial and fungal leaf spots, borers, aphids, tent caterpillar, and scale. The bark is thin and easily damaged by mowers and string trimmers. Reduce chances of disease and insects by keeping trees healthy with irrigation in an extended drought and regular fertilizer applications.
You should also check to see if your roots are girdling around the base of the tree. This may be the cause of the suckers emerging around the base of the tree. Unfortunately, there is not much that can be done about girdling because it is the result of improper planting. You should prune the suckers as they appear.
You should also be on the look-out for tent caterpillars that emerge about this time of year. They should be removed and burned as soon as they appear to limit damage to the tree.
I have included a link to a Penn State fact sheet that provided additional information on and treatment options on fruit and ornamental cherry trees. http://extension.psu.edu/pests/plant-diseases/all-fact-sheets/cherry-diseases
Regarding the sap you mention, Cherry trees can exude sap/gum due to any type of stress such as pruning cuts, wounds, cankers, or possible insect pests such as the peach tree borer. There are no chemical controls, and this pest is very common on weeping cherry trees. Without identification or an image of the tree, this is a best guess.
Here is some further information about them: http://extension.umd.edu/hgic/shade-tree-borers
Once borers are inside the tree, they cannot be controlled. They will grow into adults, emerge, and fly away. Keep your tree healthy so they don't reinfest the tree by laying more eggs on it.
Borers attack stressed trees. Keep the tree in good health through proper pruning, planting, watering, and pest management. Keep a ring of mulch around the base of the tree to conserve moisture and keep weeds down. Make sure mulch is not thicker than two inches and keep mulch away from the base of the trunk.
Mulch piled on tree trunks provides a place for the peach tree borer adult (a moth) to lay its eggs. You should be able to see the roots flaring out from the base of the trunk--the base flare should not be covered with mulch or soil.
Thank you for your question.