Has no effect unless your offshore host is slow. Some of Google's algorithm was leaked and website speed does have a factor in your rankings.
Jeez, this is almost all wrong :p, but no worries, no harm done.
Offshore hosting is one SEO factor, but a tiny one. You can get around that with Cloudfare and a proper configuration to avoid duplicate content. However, it isn't that big and you can definitly rank well without it.
Page Speed is indeed a big SEO factors that you need to tweak the best you can (image compression, no image resizing, gzip, defer JS script, remove white space and comments in html/ php files and so on.. )
Google's algorithm was never leaked. It is like the coca cola formula, they are paying the wage earners very good to avoid any leak. However, like the coca cola formula, it is studied by experts thanks to labs website. I have a few on my own which are there to identify the different factors and their weight in the algo.
To go back on your main question, the IP of your shared host is a crucial factor too at least its authority. If you are hosted near a thousands of porn websites, link farms / networks, spammy websites and so on , your IP gets dirty and your website will suffer from it.
That's why a dedicated IP is generally a requirement if you are getting serious about SEO.
