>> We will be staying for 2 days at Lucerne during the second week of April. With the cogwheel train to Mt Pilatus not operating at that time
You can go up and back on the cableway from Kriens. It will still be a great trip if the weather is clear.
>> Also, can anyone suggest how to modify this itinerary to include a couple more charming villages ?
Vitznau and Weggis are quite charming, and can be reached by boat. Going further afield, especially if you have to incorporate bus rides to get there, will probably take up too much time given the limited time you have available.
>> Will it be most economical to buy tickets for each section of the journey on the SBB website using our Half Fare Card ?
The tickets will cost the same whether you buy them on the SBB website, or from a machine or staffed ticket office on the day of travel. I would not buy in advance in case you change your mind about which day you go where (e.g. due to weather etc).
>> We are also thinking of visiting Bern for half a day (or until afternoon). At Bern we would like to walk through the old town and the river Aare at some nice spot, see the clock tower ... Is this doable by walking or we need to use public transport ? What would be the best way to travel within Bern based on our plan ?
Bern Altstadt (old town) is reasonably compact and very walkable, and also has trams which can come in handy if you get tired or want a quick trip back to the station at the end of the day.
Tickets for Bern public transport can be bought at machines and in the LIBERO shop at Bern main station. You might like to call in at the Tourist Information at Bern main station to ask for advice about the best ticket to get.
>> We would also like to visit a vantage point to enjoy a nice view of Bern. What would that be - the rose garden or there is a better place ?
There is information here about panoramic views in Bern: