How btree is stored on disc?

it all depends on DBMS you use. If you wanna know how it is implemented in MS SQL Server, things to read about are:

• Pages (I guess they are in almost all modern DBMS's) - in SQL Server they are 8Kb. Database file is composed from pages.
• Extents - logical groups of 8 continous pages
• (S)GAM - (Shared) Global Allocation Map. Bitmap containing information about free and occupied extents. This is one of the first pages on database file.
• IAM - Index Allocation Map. You can find out which index/heap is stored in which extents. Having this information, you can get place in the file where index/heap is stored.

Using IAM and GAM (or SGAM) you can split page - just move part of the page (which is supposed to be overflowed) to the another page on file.

IAM and GAM are also answers to your first question.

Most of these names are taken from MS SQL Server but I'm pretty sure, that in other DBMS's it is solved quite similar.

Hope it helps.

My recommendation is to have a look at the book Database System Implementation"

Chapter 2 "data storage" and chapter 3 "representing data elements" wil give you some hints about this problem.

Chapter 4 index structures has a section on Btrees

It's the best source of information I have found so far on this topic.