Hi,
1. yes. certificate can be use for public domain too
2. will not trust, to make it trusted, users need to install CA certificate from the CA server which created the self signed certificate.
3. since self signed certificate's CA certificate is not publicly advertised, web browsers cannot identify the trustworthiness from self signed certs. you need to install CA certificate from server to enforce web browsers to trust them.
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3.
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Good luck
KB