Hi cloudcatcher, and welcome to MySwissAlps!
Most of your plans look good!
>> Day 3- Take the train to Luzern. Begin Lake Lucerne day trip (boat, cogwheel train, cable car). Return to Engelberg.
I’m not sure what you mean: is ít a trip to Pilatus? (Boat to Alpnachstad, cogwheel train to Pilatus, cable car and gondola to Kriens, bus to Luzern? )
If you do mean Pilatus, it won’t be possible, as the cogwheel train will be closed for the winter. You can go up and back by cable car from Kriens though, if the weather is suitable.
You will probably also be able to find some time to see the Chapel Bridge and stroll through the lovely old town, as well as take a boat trip on the Vierwaldstättersee (Lake Lucerne).
Boat timetables are here:
At present, timetables are only valid up until 9 December, as a new timetable comes into operation for trains, buses, boats and cable cars on 10 December. The new timetable will be released in mid-October. In the meantime, the current December timetable up until 9 December will give you a good idea of how frequent the boat trips are at that time of year, and where they go.
It would be a good idea to plan some inclement weather options as well, as there is no point going up to mountain tops if they are in the clouds; and if the weather is too bad, cable cars will be shut down for safety reasons.
This page will give you some ideas for alternative activities:
You might also enjoy a visit to the Christmas Markets:
1. Lozärner Wiehnachtsmärt
This one is at Franziskanerplatz in Luzern (near the river, on the same side as the railway station)
Opening hours: 11:00 – 20:00 (food stands until 21:00)
2. Lucerne Christkindlimarkt
Lower ground floor of Luzern railway station (quite a small market)
Opening hours: 10:00 – 21:00