Yes. Consider the countable set $\mathbb{Q}^2$ and for each $x\in\mathbb{R}$ the subset $\{(a,b)\in\mathbb{Q}^2;a\leq x,b\leq-x\}$.

Edit: See some easier examples in the comment below by bof. There are more examples of uncountable almost disjoint families (that is, uncountable antichains where the intersection between any two elements in the antichain is finite) in [this MSE question].

Yes. Consider the countable set $\mathbb{Q}^2$ and for each $x\in\mathbb{R}$ the subset $\{(a,b)\in\mathbb{Q}^2;a\leq x,b\leq-x\}$.

Edit: See some easier examples in the comment below by bof. There are more examples of uncountable almost disjoint families (that is, uncountable antichains where the intersection between any two elements in the antichain is finite) in this MSE question.

Yes. Consider the countable set $\mathbb{Q}^2$ and for each $x\in\mathbb{R}$ the subset $\{(a,b)\in\mathbb{Q}^2;a\leq x,b\leq-x\}$.

Yes. Consider the countable set $\mathbb{Q}^2$ and for each $x\in\mathbb{R}$ the subset $\{(a,b)\in\mathbb{Q}^2;a\leq x,b\leq-x\}$.

Edit: See some easier examples in the comment below by bof. There are more examples of uncountable almost disjoint families (that is, uncountable antichains where the intersection between any two elements in the antichain is finite) in [this MSE question].