"forward" natural deduction vs "backward" natural deduction

There's nothing weird here, from your reference it states about backwards deduction feature of sequent calculus:

In order to show A ∨ B ⊢ A, B is true, you need to show A ⊢ A, B is true and B ⊢ A, B is true." Notice that in both of the subgoals, there no longer is a disjunction; in sequent calculus, we use backwards deduction to get rid of logical operators until we have atomic clauses.

So the author is emphasizing how to use sequent calculus inference rules to get rid of logical connectives in your goal $\Gamma, A , B \vdash A \wedge B, \Delta$ which contains a conjunction. Recall in SC we can have some admissible/derivable rules, but in your above case you can simply view the whole $\Gamma, A , B$ on the left of the turnstile as a multiset $\Gamma'$, then this is just the conventional $\land$ R rule.

As for contradiction on the left of turstile, it's explosion, so any $\Delta$ simply follows...