Submitting apps containing ports of chromium browser or webkit engine in the app store A small doubt here:

I am building an app which will contain either a custom, modified version of Chromium or a modified version of the webkit rendering engine. 

Does Apple have a policy of not allowing Chromium/webkit into the apps (I am guessing Apple prefers only Webcore - https://www.webkit.org/blog/114/webcore-rendering-i-the-basics/)? Is there an alternative if yes?
WebKit is in fact the only rendering engine allowed into the App Store, but you have to use Apple's preinstalled version, not your own. Relevant bit:

App Store Review Guidelines §2.17
Apps that browse the web must use the iOS WebKit framework and WebKit Javascript

So it's possible that, if it doesn't qualify as “browsing the web”, you have a chance at getting past that. Some of the most important parts of WebKit on iPhone are also closed source.
WebCore, as part of Apple's WebKit, is accepted, but it does not appear possible with these guidelines to modify it, and I think you still have to go through the WebKit framework. Plenty of people use JavaScript with JSCore in the default WebKit.
The choice to disallow all other web frameworks is honestly a bit odd, but it looks like you might have to do something besides that; it really depends on what you want to change or add from Chromium/WebKit.