Does debt include interest owed? If I calculate debt to gdp ratios, do I include the interest payment on the debt as part of the debt?
Is debt defined as a recursive sequence? 
Does current debt= interest on debt + previous debt?
Depending on your data source, the debt level will either be the face value of debt, or its market value. The usual choice is face value. The debt level is a stock.
Interest payments are an expense that will show up in the deficit, which is a flow variable.
If nothing else happens, then new debt = old debt + interest. More generally, (new debt) = (old debt) + (fiscal deficit) is the simplified relationship. (Note that this will be an approximation to real world data, as it is possible for some transactions to affect the debt level, but are not part of the definition of the fiscal deficit, and vice-versa.)