Do cancerous cells from tumors float around in the blood and show up as some kind of abnormality on blood test?

Yes, no good test. Cancerous cells do circulate in blood. There is active research going on to make medical use of this fact. However, despite repeated claims by various parties, there are no reliable, easy tests to detect these cells for routine diagnostic purposes.
It depends. Cancerous cells from tumors do travel in the blood and show up in the form of metastatic disease. Many solid tumors behave like this. But they do not appear directly in any form of blood test. Othe markers, like tumor markers and abnormal lab values can be used to determine if tumors have been completely removed or have recurred. Blood smears for leukemia, and endocrine tests are examples.