Ja, de tuin achter het kasteel en de Gloriette achterin de tuin kan gratis bezocht worden.

I couldn't understand the language written.

Yes without problem. Gardens are open and free

Depending when you go. We went February this year and did the Grand tour of 40 rooms (costing €17.50) which I thoroughly recommend, but there is a longer tour of about 60 rooms or thereabouts. Quite a few things were not open when we were there such as the maze and the zoo. The gardens were not in bloom either. I would imagine that you could spend the day there if you were planning to do all the attractions but in our case, a half day was enough. You can use the Blue line hop on hop off bus which drops you right outside the palace and it runs every 20 minutes. Hope you have a fabulous time. We certainly did.

Not sure if there are still blooms in October, but even in May, when we were there, not too many can be seen. The greenery and beautiful trees are still worth a visit, we think! Have a good visit!

Don't think you need a ticket for the places outside the palace. There is an entrance at the right side when you face the palace. Everyone just walks in... even locals go there to jog on a regular basis. Don't believe the website who ask you to buy tickets. No one checks and there are actually multiple entrances to the palace gardens.

no

depends on the kind of spring obviously, but I would shoot for early May!

you can visit the garden with unlimited time for free. you don't need a ticket for the palace to visit the gardens.

Garden entrance is free. So you can walk to Zoo gates and buy tickets there.