Strange things with Avg_row_length
There are two distinct conditions you need to be aware of
CONDITION #1 : Fragmentation
It is obvious the table has experienced lots of INSERTs, UPDATEs, and DELETEs (or non-SELECT DML). Even loading data sorted by PRIMARY KEY order will make index pages lopsided, at least 45% sparse.
I have posts about this phenomenon before

Aug 03, 2011 : When should I rebuild indexes?
Nov 26, 2012 : Does the Int type primary key insert order affect query performance in MySQL?
Oct 26, 2012 : How badly does innodb fragment in the face of somewhat out-of-order insertions? (I discuss AVL Trees Here)

Thus, data pages shrinking from 8733835264 (8.13G) to 601882624 (574M) and shrinking index pages from 4875419648 (4.54G) to (~443M) is not alarming. This will improve read performance.
CONDITION #2 : InnoDB Row Count
InnoDB is notorious for INFORMATION_SCHEMA.TABLES having spurious counts. MyISAM tables give accurate row counts in the table_rows column of INFORMATION_SCHEMA.TABLES.
InnoDB likes to do dives into the index pages in order to guess the number the rows in the table.
I have discussed this feature before as well

Feb 23, 2013 : How does mysql deal with queries that touches myisam and innodb tables? (See ASPECT #3)
Aug 04, 2011 : Optimizing InnoDB default settings
Jun 21, 2011 : From where does the MySQL Query Optimizer read index statistics?

If you want an accurate count of an InnoDB table, you must bite the bullet and run
SELECT COUNT(1) FROM calls_old; 

which will produce the same count every time.
Running this:
SELECT table_rows FROM information_schema.tables
WHERE table_name = 'calls_old';

never gives you the same value each time.
In fact, if you run
SHOW INDEXES FROM calls_old;

look at the cardinality column. It will produce different numbers each time. You would have to lock it down by setting innodb_stats_on_metadata to 0. Even if you did that, the stats would become stale over time through normal non-SELECT DML just like a MyISAM table. Running ANALYZE TABLE calls_old with innodb_stats_on_metadata set to 0 will fix the index statistics.
Having innodb_stats_on_metadata remain at 1 will make ANALYZE TABLE calls_old  completely useless. (See my post From where does the MySQL Query Optimizer read index statistics?).
Since the table is called calls_old, I can surmise that you do not need to adjust any other options since it may be an archive table as the name suggests.
EPILOGUE
The value of Avg_row_length is a good indicator that you should defragment the table. When you see an InnoDB table growing that much, you could just run
ALTER TABLE calls_old ENGINE=InnoDB;

to shrink that table.
Thus, the behavior you are seeing is driven by the two conditions I just discussed.
SUGGESTION
If you want to verify the contents, you should these two queries:

SELECT COUNT(1) FROM calls_old;
CHECKSUM TABLE calls_old;

before and after dumping/importing.