Not sure and don't know who to ask. I have a feeling that they are available most of the year. Not sure what they cost....but doubt that it is very much as they are very popular. Not that there is a nice children's play area in the park (not sure of the cost there either).

You can only visit the building with a government official invitation or during a small window of time when all government buildings are opened to the general public. Regardless. FREE. The beautiful gardens are free.

Jaff66 is correct. It is the blue line that comes from Charles de Gaulle, goes to Gare de Nord and then Gare de l'est and down to Luxembourg. The direction to look for is Robinson or Saint Remy les Chevreuse (it splits at the end so not sure which it says on the train). Bonnes Vacances!

The garden is free, there is no charge to walk through the gardens. The actual Palace is the offices of the French Senate, I do not think you can access the building, but we did not go into the building.