Yes. Thrombocytopenia, or low platelet count, is commonly caused by chemotherapy.
Yes. Supression to bone marrow resulting in anemia, low white blood cells, low platelet (Rituxan (rituximab)) - is one of the most common possible adverse events of chemotherapy= severity of problem will depend on the type of chemo. Some biological therapy may or may not cause thrombocytopenia.
Yes. Certain chemotherapy drugs have a higher likelihood of thrombocytopenia.
Yep. All blood cell lines (including platelets) can be affected by chemotherapy, some chemos more than others. Your doctor will check your CBC regularly for this. If you notice undue bleeding, though, make sure to let him/her know.