Fix resource issue on Wordpress site and move site to a new host

General information for the website: Personal blog
Description of requirements/features: Things I need doing:

1. The site has a resource issue with the DB being too big. This may be because of malware in some posts or another reason. I need this fixing. I don't want to opt for a large server resource with the hosting as my site should not be so big.

2. A new header uploading to the blog via the Header option under Appearance (not in any other way) - I will supply the header.

3. Remove any other instance of my blog eg MYSITE/cms if this is not needed as it keeps backing up this site (this was the dev site I believe?).

4. Move to a new host which is all signed up for already.

There may be one or two other small things - tbc.