Certainly, here's how you would create the tables and set up the relationships using SQL commands:

```sql
-- Creating the Borrower table
CREATE TABLE Borrower (
    borrowerID INT PRIMARY KEY,
    borrowerName VARCHAR(255),
    borrowerType VARCHAR(50)
);

-- Creating the Book table
CREATE TABLE Book (
    bookID INT PRIMARY KEY,
    bookTitle VARCHAR(255),
    Author VARCHAR(255),
    Price DECIMAL(10, 2)
);

-- Creating the Rented table
CREATE TABLE Rented (
    Bor_ID INT,
    Boo_ID INT,
    Date DATE,
    PRIMARY KEY (Bor_ID, Boo_ID),
    FOREIGN KEY (Bor_ID) REFERENCES Borrower(borrowerID),
    FOREIGN KEY (Boo_ID) REFERENCES Book(bookID)
);
```

Once the tables are created, you can add valid data using `INSERT INTO` statements:

```sql
-- Adding valid data to the Borrower table
INSERT INTO Borrower (borrowerID, borrowerName, borrowerType)
VALUES (1, 'John Doe', 'Student'),
       (2, 'Jane Smith', 'Faculty');

-- Adding valid data to the Book table
INSERT INTO Book (bookID, bookTitle, Author, Price)
VALUES (101, 'Introduction to Programming', 'Alice Johnson', 29.99),
       (102, 'Data Structures and Algorithms', 'Bob Anderson', 39.99);

-- Adding valid data to the Rented table
INSERT INTO Rented (Bor_ID, Boo_ID, Date)
VALUES (1, 101, '2023-08-01'),
       (2, 102, '2023-07-15');
```

For testing with invalid data, you can try inserting data that violates constraints, such as using a non-existent borrower ID or book ID as a foreign key in the `Rented` table, or inserting duplicate primary key values in the `Rented` table. SQL systems will generally throw errors for such cases, indicating that incorrect data is being discarded.

The `Date` datatype is used for storing dates. You can use the format 'YYYY-MM-DD' for inserting dates.

As always, keep in mind that the actual SQL syntax might slightly vary depending on the database management system you're using.