drush sqlc import causing Mysql 1045 (28000) Error Acces. How to resolve it?
To reduce the possibility that the size of your database is what is causing your problem, I would try to use the Backup and Migrate module.

In doing so, you can exclude all sorts of tables (which might reduce the size tremendously), similar to what I described in my answer to the question about "How to get an import of a very large Drupal database working?".

PS: my answer to your 2nd question (= Should I update the live site to 7.41 before migrating it?) is No, because you should not have to update a live site to create a development environment. If you have a need for them to be at the same level (to start from), then rather do the opposite: have your local site start from 7.39, then sync your database, and if needed (only) then upgrade your local site to 7.41. You'd get the same result, without touching your live site, right?