Noob

Use a whois service to find out where the domain is registered. You then contact the registrar and get a login to your control panel. You can then set up redirects from the old site to new - this will help retain all your ranking.
If you need help with this let me know - no charge for the service as it's a simple thing to do once you know how to do it.

Hi,
I'm no WordPress expert, many around here are and would know platform specific issues but i do know a thing or two about general speed and stuff. The problems reported by Pingdom are also reflected by Google PageSpeed Insights.
The main concept I found useful is to divide your css and js into 'critical' which is required to shape the page and 'action based' which is js and css which are used on an interaction, e.g. validation js is not required to view the form it's only used to validate a submission or a lightbox gallery is only required when a visitor clicks to open.
Once done you inline critical css in the head with noscript to the action based css and then load the rest with js in the foot of the page. A simple but handy script I found for this is as follows;
<script type="text/javascript"> function loadjscssfile(filename, filetype){ if(filetype=="js"){ var fileref=document.createElement('script'); fileref.setAttribute("type","text/javascript"); fileref.setAttribute("src", filename); fileref.setAttribute("charset", "utf-8"); } if(filetype=="css"){ var fileref=document.createElement('link'); fileref.setAttribute("rel", "stylesheet"); fileref.setAttribute("type", "text/css"); fileref.setAttribute("href", filename); } if(typeof fileref!="undefined") { document.getElementsByTagName("head")[0].appendChild(fileref); } } loadjscssfile("/js/action.js", "js"); loadjscssfile("/css/action.css", "css"); </script> With regards to the initial load, could be the server itself being slow or it could be something WordPress is doing before it returns the page. WordPress experts can help if it's that but a good test for a server is to upload a simple html only page and then speed test it, if that is fine then try again with a simple php page with something simple like;
<?php echo $_SERVER['REQUEST_URI']; ?> This should see if it's the server speed processing php that is at fault. 
As for the image, no idea mate. Normally if you delete something it's gone, not sure if WordPress has some safeguard which modifies delete permissions.
Hope some of this helps