A convertible car is fun to drive. Isaac’s car is not a convertible. Therefore, Isaac’s car is not fun to drive.
Letting $C(x)$ be "$x$ is a convertible car" and $F(x)$ as "$x$ is fun to drive", the first premise of the above argument is that $\forall x ( C(x) \to F(x) ).$ Since we know that ~$C(x)$ is true, we cannot say that the argument's conclusion ~$F(x)$ is true.
A dog can bark. Arthur's pet is not a dog. Therefore, Arthur's pet cannot bark.
However, in this argument, we cannot make a similar inference as in the previous argument. Logically, knowing that only dogs can bark, we can arrive at this argument's conclusion. How do I use Mathematics to represent this thinking? How would a mathematical system or a computer would identify the correct conclusion?
No, you know that ~$C(c),$ where $c$ specifically stands for Isaac's car, is true. ~$C(x)$ may or may not be true, depending on which car $x$ stands for.
First, let's denote Arthur's pet and "$x$ is a dog" and "$x$ can bark" by $p$ and $Dx$ and $Bx,$ respectively.
By feeding it the premise $\forall y(By\to Dy),$ which was implicit to you but missing from the second given argument. That is, the argument that you actually want is $$\forall x (Dx\to Bx)\quad\text{and}\quad \lnot Dp\quad\text{and}\quad \color\red{\forall y(By\to Dy)},\quad\text{therefore}\quad\lnot Bp,$$ which, as you've suggested, unlike the first argument, is valid. In other words, the sentence $$\bigg(\forall x (Dx\to Bx)\quad\land\quad \lnot Dp\quad\land\quad \forall y(By\to Dy)\bigg)\to\lnot Bp$$ is logically valid.
This new argument is actually unsound though: sea lions also can bark, so the premise "only dogs can bark" is false!