Fully Homomorphic Encryption over the Integers - Runtime Question
On the first question

The public key of DGHV's SHE scheme consists of $\tau+1$ $\gamma$-bit numbers, that is, $pk = (x_0,\dots,x_\tau)$, where $x_i$ is chosen from the distribution $\mathcal{D}_{\gamma,\rho}(p)$ over $[0,2^{\gamma})$.
Therefore, the length of $pk$ is $O(\tau \gamma)$, which is $\tilde{O}(\lambda^{10})$ if we adopt the parameters $\gamma = \tilde{O}(\lambda^5)$ and $\tau = \gamma + \lambda$ as the authors said.

In order to encrypt a bit $m$, we sample a random subset $S$ and a "relatively small" integer $r$, and compute $c = m + 2r + 2 \sum_{i \in S} x_i \bmod{x_0}$.
Here, we have to read the whole of the public key and the main task is adding $x_i$ randomly. Hence, the complexity of encryption is $\tilde{O}(\lambda^{10})$, which is the result of $\tau$-time addition of $\gamma$-bit integers.

In order to multiply two fresh ciphertexts, which are $\gamma$-bit integers,
we require about $O(\gamma^2)$ bit operations. (By using more sophisticated multiplication, it is reduced to $\tilde{O}(\gamma)$.)

As you said, the time of evaluating a circuit and encrypted data mainly depends on the depth of the circuit.
Getting worse, we are often required to adopt the FHE scheme and "refresh" intermediate ciphertexts.

On the second question

As an example, we take $\rho = \log^{2}(\lambda)$.
This results in $2^{2\rho} = \lambda^{2 \log(\lambda)}$, which is asymptotically greater than any polynomial of $\lambda$.
Hence, asymptotically speaking, the attack is not faster than the runtime of the scheme.
As you thought, if we take $\rho$ is very close to $\log(\lambda)$, the scheme is unsafe for small $\lambda$.