The first issue is your hosting company fault. The underlying library that performs the authentication that you are actually connecting to wordpress.org is probably missing some related configuration to be able to complete correctly the authentication.
The solution they gave you is not secure and while I don’t believe anything bad will happen because of that it is a good reason to start suspecting the security practices of the company and maybe look for a better one.
As fr the second part, you didn’t give/have the right permission to the upgrade process to be able to write to the directory where your wordpress is installed. If you don’t understand file permission issues then you ask the support to check them for you.