Answers to the common questions. Anything else: psppublishingpress@gmail.com.
macOS asks before any app reads the clipboard. Open System Settings › Privacy & Security › Paste from Other Apps and set Rekeep to Always Allow. Also check that capture isn’t paused (menu bar icon › Resume Capture) and that the app you copied from isn’t excluded (Settings › Privacy).
Pasting for you needs Accessibility permission: System Settings › Privacy & Security › Accessibility › turn on Rekeep. Without it, Rekeep copies the clip and you press ⌘V.
iOS doesn’t let any app watch the clipboard in the background. When you open Rekeep after copying, tap Paste (no permission prompt). From any app you can also use the Share button and choose Rekeep.
Settings › General › Keyboard › Keyboards › Add New Keyboard › Rekeep, then tap Rekeep and turn on Allow Full Access so it can show your saved clips. Switch to it with the globe key. It inserts text clips; for images and files, copy them in the app.
Nothing. Every feature is free, with no ads, no subscription, and no account.
Not in version 1.0 — each device keeps its own library, privately. On Mac you can export your library to a folder and import it on another Mac.