How can I determine which fields are specified as NOT NULL within the database?

I'm not sure if this is the best approach but I created an extension method for the DbContext class that takes a string name of a data table and then queries the sys tables for this meta information. Specifically, I created the following 2 classes and extension method.

TableSchema

This is the high-level, table class used to store pertinent schema details:

public class TableSchema
{
    public string Database {  get; internal set; }
    public string TableName { get; internal set; }
    public List<ColumnSchema> Columns { get; internal set; }
}

ColumnSchema

Much like TableSchema, this is the class which will contain all schema related details for each column.

public class ColumnSchema
{
    public string ColumnName { get; internal set; }
    public int ColumnPosition { get; internal set; }
    public string Collation { get; internal set; }
    public string TypeName { get; internal set; }
    public short Size { get; internal set; }
    public byte Precision { get; internal set; }
    public byte Scale { get; internal set; }
    internal int _PK { get; set; }
    public bool IsIdentity { get; internal set; }
    public bool IsNullable { get; internal set; }

    public bool IsPrimaryKey
    {
        get { return _PK == 1; }
    }
}

The Extension method (GetDbTableSchema)

This method extends the DbContext class. This makes acquiring the underlying table details as simple as passing a name into a method, right off of your instantiated context.

public static class DbContextExtensions
{
    public static TableSchema GetDbTableSchema(this DbContext ctx, string tableName)
    {
        string qry = string.Format(
    @"SELECT * FROM (SELECT DISTINCT 
        c.name AS ColumnName, 
        c.column_id AS ColumnPosition,
        ty.name AS TypeName,
        c.max_length AS Size,
        c.precision AS Precision,
        c.scale AS Scale,
        CASE WHEN ic.column_id IS NOT NULL THEN 1 ELSE 0 END AS [_PK], 
        c.is_identity AS [IsIdentity],
        c.is_nullable AS [IsNullable]
    FROM sys.columns c 
        INNER JOIN sys.tables t ON c.object_id = t.object_id 
        INNER JOIN sys.types ty ON c.system_type_id = ty.system_type_id
        LEFT OUTER JOIN sys.indexes i ON c.object_id = i.object_id AND i.is_primary_key = 1
            LEFT OUTER JOIN sys.index_columns ic ON i.object_id = ic.object_id 
                AND i.index_id = ic.index_id 
                AND c.column_id = ic.column_id
    WHERE t.name = '{0}') t
    ORDER BY _PK DESC, ColumnPosition", tableName);", tableName);
        return new TableSchema
        {
            Columns = ctx.Database.SqlQuery<ColumnSchema>(qry).ToList(),
            Database = ctx.Database.Connection.Database,
            TableName = tableName
        };
    }
}

Usage is very simple. Assuming you have the name of the data table, pass it into your context.

using (var ctx = new MyEntityContext()
{
    TableSchema ts = ctx.GetDbTableSchema("MyTable");

    foreach (ColumnSchema cs in ts.Columns)
    {
        Debug.WriteLine("Column: {0}, {1}", cs.ColumnName, cs.IsNullable ? "NULL" : "NOT NULL");
    }
}