Does the migration required any user-side action?
I had a seed of ARK wallet written somewhere - should I perform any action, or my funds will be still accessible with this old seed?
Nothing needs to be done from user, user shouldn't experience any difference when interacting with Ark blockchain all of this happened in the "back" with the ones who are running the network (delegates).
And what are the actual reason for migration? Understandable for common people

(I am sorry, but I haven't really understood the last update in details.)
Is it easier to develop and to program on typewrite or something like that?
I touched on the why's in the blog post:
TypeScript is also a much stricter language than JavaScript and lets you spot mistakes that could otherwise go unnoticed. It is more convenient to use and allows for quicker error debugging. For developers, very little will change as the syntax is almost the same as JavaScript (Typescript is a superset of JavaScript). It enables JavaScript developers to use highly-productive development tools and practices, like static checking and code refactoring, when developing applications. TypeScript makes perfect sense for growing projects that need to scale and have a variety of contributing developers.