Yes. Batch PDF processor works inside your browser. Your files are not uploaded anywhere, so nobody else can see them, and there are no waiting queues.
No. It works in any modern browser on a computer or phone.
Batch PDF processor is one of 12 Premium tools. You can pay ₹5 for a single document, or get a Premium pass from ₹49 that also removes ads, paid once by UPI, card or net banking. The other 39 tools on DoPDFNow are free.