Yes and a dedicated server wouldn't make it faster, because your problem is not the server. Most of all requests come from external hosts like Google, Facebook and so on. Kick them off and your shop goes 3-times faster, atleast!!!
For developing I have the same. On this server I have around 10 different shops running. The biggest one is a Shopware shop, similar to Magento and this Shop has around 50,000 products. It is fast, really fast, so I think you don't need a VPS.
I think you are a dreamer? ;)
There is no provider in this universe who allows a higher max_execution_timeout higher than 300s on shared hosting. The same with memory limit! If so nobody would pay for a dedicated server!
In a certain respect you are certainly right with your accusations, but the problem has to be put into perspective. The cache extension for OC actually works perfectly as long as you don't use Journal 3 and you don't use the crawler, because Journal 3 has a device detection and generates...
There is no option or parameter in URL to display the mobile view on desktop. Which code (HTML, css,js,....) is used depends on User Agent, so you must request your page with a mobile device to get the code and the view for this device (User Agent).
Don't know which plugin you talking about, but to simulate Mobile View on desktop open Chrome or Firefox and press Strg + Shift + M on your keyboard and you will see what you want to see, so there is no need for a online-tool.
Again, as long as you have cache varies in .htaccess defined warmup with crawler will not work. The UA of the crawler is fix and doesn't match with cache varies in .htaccess.