Is RSA in a ECB-like-mode safe for bulk encryption?
Well, reusing a key isn't a problem; after all, RSA keys are generally used many times.

However, if you fix the padding, there does exist one other potential problem; message malleability.

To example, suppose Alice sends two messages to Bob, $X_1, X_2$ and $Y_1, Y_2$.  To send these, Alice actually sends:

$E(X_1), E(X_2)$

$E(Y_1), E(Y_2)$

Now, Eve can't modify each individual block (RSA with proper padding prevents that); what she can do is mix-and-match blocks, as so:

$E(X_1), E(Y_2)$

This would decrypt at $X_1, Y_2$, which might not be what Alice and Bob whats.

If you are silly enough to use RSA encryption in this manner, you do need something that binds the blocks together (or alternatively does some sort of integrity checking on the entire message).

As for whether people ever do this in practice, well, no they don't -- not because of security, but because it's so inefficient; using the RSA algorithm to communicate a symmetric key, and using that symmetric key to encrypt the message is far more efficient.