does automatic checkpoint flushes dirty pages which are related to un-committed transactions to MDF File in SQL Server
does it also flushes dirty pages which are related to un-committed
  transactions into data file or it only flushes dirty pages related to
  committed transactions?


Yes of course, it flushes all the dirty pages:


  When a checkpoint operation occurs, no matter how it’s triggered (for
  instance through a manual CHECKPOINT, from a database or differential
  backup, or automatically) the same set of operations occurs:
  
  
  All dirty data file pages for the database are written to disk (all    pages that have changed in memory since they were read from disk or
  since the last checkpoint), regardless of the state of the
  transaction that made the change.
  Before a page is written to disk, all log records up to and including    the most recent log record describing a change to that
  page are    written to disk (yes, log records can be cached in memory
  too). This    guarantees recovery can work and is called write-ahead
  logging. Log    records are written to the log sequentially, and log
  records from    multiple transactions will be interspersed in the log.
  The log cannot    be selectively written to disk, so writing a dirty
  page to disk that    only has a single log record affecting it may
  mean writing many more    previous log records to disk as well.
  Log records describing the checkpoint are generated.
  The LSN of the checkpoint is recorded in the database boot page in    the dbi_checkptLSN field (see Search Engine Q&amp;A #20: Boot pages, and
  boot page corruption).
  If in the SIMPLE recovery model, the VLFs in the log are checked to    see whether they can be marked inactive (called clearing or
  truncating the log – both of which are terrible misnomers, as nothing 
  is either physically cleared or truncated).
  


How do checkpoints work and what gets logged by P. Randal


  I am interested to how SQL Server does recovery in case of system
  failure and how it depends on CHECKPOINT and are there any other
  factors on which DB recovery depends?
  
  During crash recovery, the mechanism is more complicated. The fact
  that database pages are not written to disk when a transaction commits
  means that there is no guarantee that the set of database pages on
  disk accurately reflects the set of changes described in the
  transaction log—either for committed or uncommitted transactions.
  However, there is one final piece of the puzzle that I haven't
  mentioned yet—all database pages have a field in their page header (a
  96-byte portion of the 8192-byte page that contains metadata about the
  page) that contains the LSN of the last log record that affected the
  page. This allows the recovery system to decide what to do about a
  particular log record that it must recover:



For a log record from a committed transaction where the database page


  has an LSN equal to or greater than the LSN of the log record, nothing
  needs to be done. The effect of the log record has already been
  persisted on the page on disk.

For a log record from a committed


  transaction where the database page has an LSN less than the LSN of
  the log record, the log record must be redone to ensure the
  transaction effects are persisted.

For a log record from an


  uncommitted transaction where the database page has an LSN equal to or
  greater than the LSN of the log record, the log record must be undone
  to ensure the transaction effects are not persisted.

For a log record


  from an uncommitted transaction where the database page has an LSN
  less than the LSN of the log record, nothing needs to be done.




  The
  effect of the log record was not persisted on the page on disk and as
  such does not need to be undone. Crash recovery reads through the
  transaction log and ensures that all effects of all committed
  transactions are persisted in the database, and all effects of all
  uncommitted transactions are not persisted in the database—the REDO
  and UNDO phases, respectively. Once crash recovery completes, the
  database is transactionally consistent and available for use.


Understanding Logging and Recovery in SQL Server by P.Randal  


  Does recovery process depends on Recovery model as well?


No, it does not.
Database should be consistent in any recovery model after crash recovery.