How can an adversary use Mac(.) with a new key to compute a tag although this is not a query to Mac oracle?

Question

I am having some troubles understanding the power of an adversary in attack against a MAC.

1. In chapter 4 in Introduction of Modern Cryptography by Jonathan Katz and Yehuda Lindell, specifically in the message authentication experiment ($Mac-forge_{A,\Pi}(n)$), an adversary $A$ is given access to MAC oracle $Mac_k(\cdot)$ where $k$ is previously generated by $Gen$. So my question is: Is it right that $A$ can only access to $Mac(\cdot)$ with a fixed key $k$? Can $A$ access to $Mac(\cdot)$ with a different key $k'$ in a same experiment?
2. In the paper The Power of Verification Queries in Message Authentication and Authenticated Encryption, specifically in Claim 4.2, it is stated that adversary $A$ simply runs algorithm $Mac(\cdot)$ with key $L$ and message $M$ (where $L$ is a key forged by $A$). I do not understand this part.

Question: How does $A$ run the algorithm if it is not a $Mac$ query? If possible, could you please provide an example in reality?

Answer 1

In modern cryptography it is generally assumed that algorithms are public and only the key is kept private. Thus, the adversary can compute $\operatorname{Mac}(k', m)$ for any key $k'$ and message $m$.

The oracle $\operatorname{Mac}_k(\cdot)$ in the experiment allows $A$ to additionally receive valid macs under the challenge-key $k$ for which it is supposed to forge a valid mac (for some message $m^\ast$ that it has not queried from said oracle).

The same goes for the second part of your question: the algorithm is public, so $A$ can run it on any input.