You have to ask it to Indonesian Embassy in your country thi is prohibited, entrance is free foe all people this officer take the money for him

Depending on which part of Batam you are going to, there are many ferry terminals and many different ferry companies so it is best to find out which ferry terminal is nearer to your destination then decide on which ferry terminal. Some ferry companies use fast boat while some use large catamarans which are more stable and comfortable. No problem returning the next day, in fact many people do day trips. However, do note that the last ferry during weekends can be quite full so when you arrive, you might want to check with the ferry company on the timing for the return trip and decide on which timing you would like to take.

Your hotel should be able to tell you the best terminal, and should be able to arrange a car to pick you up!

You should take the ferry to Vatam Center. From there you can just about get any part of the city. The ride should take about an hour.

No idea

If you go from Singapore all seaport tax is included for two ways.

You can buy from the ferry terminal. It's easy and cheap.

Weekends (Saturday and Sunday) and public holidays are always packed. However, at ferry terminal, crowd management is good and you can clear the immigration check then come out in an hour.

I have asked friends in Batam about this. They told me there is no ferry service from Batam to Lake Toba. Probably you can hire a private boat to go there instead.