Can I use the "import" tool on PHPMyAdmin multiple times?
Yes this is possible, although you need to watch out for the following:

a) Export using extended inserts so that the extended SQL syntax is used which will reduce the size of your SQL files (If you are exporting using PHPMyAdmin select both complete and extended insert options, as it creates complete inserts with extended insert syntax for large tables) 

b) Try using the GZip format which will compress the SQL files and make them smaller (depending on the hosting provider setup this may be an issue so you need to test it out with a small table) 

c) Export the tables in such an order that tables with foreign keys are imported last