Give an simple sqlite3 table (create table data (key PRIMARY KEY,value)) with key size of 256 bytes and value size of 4096 bytes, what is the limit (ignoring disk space limits) on the maximum number of rows in this sqlite3 table? Are their limits associated with OS (win32, linux or Mac)
Essentially no real limits
see http://www.sqlite.org/limits.html for details
The answer you want is right here.
Each OS you mentioned supports multiple file system types. The actual limits will be per-filesystem, not per-OS. It's difficult to summarize the constraint matrix on SO, but while some file systems impose limits on file sizes, all major OS kernels today support a file system with extremely large files.
The maximum page size of an sqlite3 db is quite large, 2^32768, although this requires some configuration. I presume an index must specify a page number but the result is likely to be that an OS or environment limit is reached first.
In SQLite3 the field size isn't fixed. The engine will commit as much space as needed for each cell.
For the file limits see this SO question:
What are the performance characteristics of sqlite with very large database files?