For question (1), the Boolean ultrapower construction provides a manner of doing forcing where the extension has new ordinals. Specifically, with the Boolean ultrapower, one has a model of set theory $M$ and a forcing notion $\mathbb{B}\in M$ (a complete Boolean algebra in $M$), with an ultrafilter $U\subset\mathbb{B}$, not necessary generic in any sense and $U\in M$ is completly fine. One forms the Boolean ultrapower of $M$ by $U$ by taking equivalence classes of $\mathbb{B}$-names in $M$ under the equivalence $$\sigma=_U\tau\iff[\![\sigma=\tau ]\!]\in U$$ and relation $$\sigma\in_U\tau\iff[\![\sigma\in\tau ]\!]\in U,$$ and this gives an elementary embedding $j:M\to {\check M}_U\subset \check M_U[G]=M^{\mathbb{B}}/U$. The interesting case occurs when $U$ is NOT $M$-generic, for it is in this case that the map $j$ is nontrivial. Equivalently, in this case the structure $\check M_U$ and forcing extension $M^{\mathbb{B}}/U$ have new ordinals not in $M$. Thus, the nontrivial instances of the Boolean ultrapower are exactly when you find new ordinals in the forcing extension.
For question (2), one example might be axiom A forcing, where in addition to the usual order $\leq$ on the partial order $\mathbb{P}$, one has supplemental orders $\leq_n$ that interact in a way allowing one to undertake fusion-like arguments. So with Axiom A forcing, one does not have merely a partial order, but really a coherent nested tower of partial orders that interact in a useful way. Axiom A forcing pre-dated and I believe led directly to the concept of proper forcing, which has been extremely important in the development of forcing.
It also seems relevant to mention in regard to question (2) that while classical forcing uses (complete) Boolean algebras, in topos theory one undertakes the forcing construction with a Heyting algebra, rather than a Boolean algebra, with the result that one gets a non-classical logic in the forcing extension. Probably some of the topos theoreticians here on MO can say more about this.
