Hi bella2020, and welcome! I think you've definitely managed to arrange an impressive travel schedule. It may be a bit too tight on some days though. Some thoughts on your travel scheme:
Day 1: renting a car is an option or course, and I understand that it may come in handy when stayin in Beatenberg. Swiss public transport is great as well though. You could easily cover all travel plans using buses and trains.
Day 2: I'd personally prefer to visit the Jungfraujoch as early in the morning as possible. On most days, skies are clearer in the morning than in the afternoon. Moreover, the sun will be in a better position for making pictures (I'm not sure if this is important to you) towards the Aletsch glacier. However, it's best to adapt your scheme to weather circumstances of that day. Just check weather forecasts the evening before or the same morning, and then decide which order of plans is best.
With or without visiting the Jungfraujoch, I would advise to get down via Grindelwald. The Männlichen cable car offers great views of the Lauterbrunnen valley, and so does the cog railway down from Kleine Scheidegg and Wengen-Lauterbrunnen. Returning via Grindelwald allows you to explore another beautiful valley.
Day 3: Both Jungfraujoch and Schilthorn are great in their own ways, providing different views and experiences. If you like visiting mountain tops, and if weather is fine, just go for the Schilthorn as well. In all, this will be quite a full day. If it's a sunny day, you may want to skip the Trümmelbach falls (although they're quite impressive) and enjoy a relaxed afternoon or evening in Beatenberg.
Day 4: This is do-able, although it will be quite a long day as well.
Day 5: I'm not sure how much time you'd like to spend in both Lucerne and Mendrisio, but this may be too much for one day. Are you planning on staying in Mendrisio for the night, or do you have to move on to Italy the same day? Anyway, you'll have to drive to Lucerne first, which takes about 1h30. The trip from Lucerne to Mendrisio by car takes another 2h15/2h30. You may want to slow down a bit along the way and take some breaks, as the views are truly beautiful. If you leave early, it's all possible, and it's a matter of personal preference too. For example: you could spend your time in Lucerne from 9am to 12am, travel onto Mendrisio and do some shopping there between 3pm/4pm and 6 pm.
Yes, even with a car, the Swiss Half Fare Card
will save you quite some money on the Jungfraujoch trip, other mountain trains and cable cars, and the boat ride.
Some bad weather alternatives are the Trümmelbach falls, the Aare gorge near Meiringen and the Beatus caves near Beatenberg. Further tips can be found on our Bernese Oberland day trip overview
I hope this helps a bit!