Hey, feel free to reach out to us personally at firstname.lastname@example.org, that said I'll try to answer your questions here:
how much are you charged for your first time?
The smallest plan is $100 per month there is no installation charge.
If you don't give your monthly pay what happens, will your forum be shutdown completely?
At worst if you are a few weeks behind we will just temporarily disable. We'll keep your data intact. Also if you decide to move to your own self-hosting (see next point) we will provide you the database backup.
Or is there any cheaper way to get NodeBB?
Well NodeBB is open source, so if you were to decide to host it yourself you're more than free to do so.