Tap the "+" button in the top right corner, enter their name and phone number. You can also include an email address or photo.

Yes. Go to Settings > Import Contacts and select which app you want to import from (e.g. Google Contacts). Follow the prompts to complete.

Yes. Select "Edit" on the main screen, then tap all of the contacts that you'd like removed. Finally, press "Delete."

If your data was backed up via iCloud/Google account before losing your device simply log into these accounts after getting a new one & sync back.