For checked-in items that are not oversized/overweight/irregular/special handling items, is bag weight routinely stored in the PNRs by any airlines?

PNRs do get forwarded and checked by authorities in some countries, but that is not standard and routinely applied to all arriving passengers in most countries. In most EU countries' airports, customs checks don't involve a reliance upon the PNR data as much as flagging down passengers with multiple large suitcases originating outside of the EU (or a subset of EU countries).