You want to be sure everything is working before you flip the big switch :)
One quick and easy way to test the new Wordpress install is to add an entry to the hosts file on your own desktop/laptop. I.e., point the live site domain to the IP of the new Wordpress setup. This way, you can test away as much as you like on your own system without impacting anyone else.
Wordpress needs the 'site address' DB setting to be right or it acts weird. If you use the hosts file trick above, then you can set it to it's 'production' setting and test it. You won't need to flip it when you go live. Whereas if you set the site up on a test domain, it's another change you need to make when you go live.
I've also come across some plugins which don't relocate well and can need to be re-installed when the site_url is changed. So, watch out for this.
On the database side of things, I suggest, as a trial, timing how long it takes to dump out the db from the old server, copy it to the new server and import it over there. Chances are, if you have a large DB, the import will be the longest step. You'll then need to take that time and add the TTL of the DNS records to it and the time it'll take you to change them to give you a total time taken for the migration. Is this time long enough that there could be significant activity on the old site? I.e., could someone have made an order on the old site that will now be missing on the new site? Normally, one way to avoid this would be to set up DB replication between old DB and new, although, a shared hosting setup probably won't let you do this. You should consider having an actual proper scheduled downtime window so you don't lose any user/customer data that's created during the migration.
As for the email, these are some things you need to check:
- What is the new server calling itself in its HELO's? Should be something like mail.yourdomain.com. There should be an A record for this and it should point to the right IP address. There should also be a corresponding PTR record.
- Is your IP in any of the common spam lists? E.g., spamhaus?
- Do you have SPF records set up for the new mailer?
Good luck! :)