Murten

Do you need the car around the city of Bern? S-Bahn train takes 35 minutes with direct service running hourly (and additional connecting departures on the alternate half hours).

Parking. The centre of Bern (as all Swiss cities) is a no driving zone, you will have to park outside the centre, which will cost, and walk in. Before you go to Bern you need to research where the car parks are and head for one of them as your destination.
You can also get by train from Murten to Bern in 41 minutes.
And don't forget to visit the bears.
Even in a smaller town like Murten, it is not a given that a hotel will have parking - always check.

2 says for the area around the lake

Which lake, do you mean the one Murten is on (Murtensee)?
I hope our stay in Switzerland includes time to see the Alps and the other parts of Switzerland: