Help with domain name, hosting, and updating SMPT records so emails are sent

General information for the website: Education
Description of requirements/features: Please see notes below
Extra services needed: Hosting, Security
Extra notes: Hi,

Due to a mixup with our current host (long story), I need to find a way to download our website in full, create a new hosting account, and update our domain servers to go to the new host.

The domain is registered with GoDaddy (which I have password and logins for) and hosting is with Host Gater (which I don't have login details for). The site is a Wordpress site, and I have login details to edit via Wordpress.

Our email for the domain is redirected through Gmail. We are having a couple of issues with emails not always sending to certain mailservers - Error messages I receive are:

550 5.7.1 Sender ID (PRA) Not Permitted

and also

Technical details of permanent failure:
Google tried to deliver your message, but it was rejected by the server for the recipient domain esl1.com.cn by qiye163mx01.mxmail.netease.com. [43.230.90.12].

The error that the other server returned was:
550 MI:SPF mx10,veCowACn_17S3DxWoc1MAA--.1S3 1446829292 http://mail.163.com/help/help_spam_16.htm?ip=209.85.215.52&hostid=mx10&time=1446829292


So, I need an internet Wizard who knows how to get things working. I'm sure it's just a quick fix, but a bit beyond me at this stage.

The final thing I need, is to do a password reset for one of the domain accounts linked with Gmail. The only way we can do this is to update DNS records, so I'm guessing the new hosting will have to be in place first?

Please quote me a fixed cost on how much you can do this for. Don't want to go into specifics on the domain here, so apologies if anything sounds too vague, but feel free to private message and we can go from there.

Kind regards,

Josh