Sampling values and perform Linear operation on the Laplacian Random Variable

Question

I am trying to sample $k$ values from a Laplacian distribution with mean $\mu$ and SD $\sigma$ in two different ways and performing simple linear operations on that of form $c_i * x_i$, where $x_i$s are the sampled values and $c_i$s are constants.

1. In the first way, I am sampling independent values $k$ times from $Lap(\mu, \sigma)$ to obtain $x_1,\cdots, x_k$ and performing $c_i * x_i$.

Question:

i) Do $x_1,\cdots, x_k$ belong to a Laplacian distribution with mean $\mu$ and SD $\sigma$?

ii) Do $c_1 * x_1,\cdots, c_k * x_k$ still belong to a Laplacian distribution? If yes, how to obtain the mean and SD of that?

2. In the second way, I am sampling $x_1,\cdots, x_k$ together from $Lap(\mu, \sigma)$ to obtain $c_i * x_i$.

Question:

i) Do $c_1 * x_1,\cdots, c_k * x_k$ still belong to a Laplacian distribution? If yes, how to obtain the mean and SD of that?

I am very much new to the statistics domain and came across a research problem that requires an understanding of the above questions. I am trying to identify which procedure out of the above two is best suited for my problem.