Trains are running more or less normally. You can check schedules/information on the SBB website.
The main highways are open and there are not "multiple road closures", just some diversions and the normal summer road works etc. Maybe that's what you are seeing? In Lucerne, there are some detours due to the flooding, but nothing that would stop you getting into the city. Unfortunately I don't know of any good traffic sites in English, but the TCS site (like the AAA of Switzerland) has fairly easy to follow maps and you can add webcams to the map if you want to look at the traffic yourself for reassurance https://www.tcs.ch/de/tools/verkehrsinfo-verkehrslage/aktuelle-lage.php
There are some flooding issues in/around Lucerne which have been affecting certain public transport routes and roads (for example the road to the Verkehrshaus museum) as well as various attractions in the region. The promenade along the lake is still flooded, and the lake ships are also not operating at the moment (currently until at least 21 July). The water level is slowly dropping and the pedestrian bridges in town were reopened this morning, but there are still some businesses closed along the riverside (not clear when they will reopen).
There is still a lot of water around, but hopefully things improve further over the next few days. We have sunny, warm weather forecast for this week....but it may rain again from this weekend, so keep an eye on things and just prepare for the likely scenario that everything is not 100% "normal" when you get here. I'm sure the tourist office can advise you about the latest service changes/closures/etc once you are in Lucerne.