1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is well-known that in $L[G],$ $0^\sharp$ does not exist, hence by Jensen's covering lemma the pair $(L, L[G])$ satisfies the covering lemma; i.e any uncountable set $X \in L[G]$ of ordinals is covered by a set $Y\in L$ of the same cardinality.
Is there a direct proof of this fact without using Jensen's covering lemma and using the properties of $P$ and the fact that we are forcing over $L$?
2-Work in $L$. Let $P$ be the Easton support product of forcing notions $Add(\kappa^+, 1), \kappa$ a singular cardinal. Then $P$ is tame, preserves cardinals and the $GCH$. Is there any tame class forcing notion $Q$ such that forcing with $P\times Q$ over $L$ collapses all cardinals into $\omega$
Remark. It is possible to define a tame and cardinal preserving class forcing notion $P$ over $L$ whose product $P\times P$ collapses all cardinals into $\omega$.
