yes, you can use TablePress already now. Also, you won't loose data, as TablePress does not influence or change anything from WP-Table Reloaded. As soon as you deactivate TablePress again, WP-Table Reloaded will resume as before.
Here are some instructions for a good start:
1. Just to be on the safe side, create a "WP-Table Reloaded Dump File" (on the "Plugin Options" screen of WP-Table Reloaded), as a complete backup.
2. If you want, you can addionally export your most important tables as CSV files, as a backup. If you are really afraid of data loss, it also can't hurt to create a full backup of your WordPress mySQL database (which you should be doing regularly anyway.)
3. Then install and activate TablePress. (Your tables will now not show on your site, but that is expected, and only temporary until you complete step 4.)
4. In TablePress, you will see a notification that WP-Table Reloaded is installed and activated. Click that large button to get to the "Import" screen of TablePress. In the lower half, you can find an import possibility. Leave everything as it is there and click "Import from WP-Table Reloaded". This will import your WP-Table Reloaded tables and their options into TablePress, as well as your "Custom CSS" (if you have any). The "Custom CSS" might need some updating, if it used special things, but likely that is not the case.
Your tables will now show again on your site, now created by TablePress, which you can recognize from the new (more fresh and modern) styling.
5. You can now work with your tables in TablePress. Changes that you make in TablePress tables will NOT influence/change/update WP-Table Reloaded tables!
6. Once you are satisfied, you should deactivate WP-Table Reloaded. However, before you do that, make sure that the "Remove upon Deactivation" checkbox on the "Plugin Options" screen of WP-Table Reloaded is NOT checked (unless you really want to remove all WP-Table Reloaded tables from the database). If you want to remove everything, you can also click the "Uninstall" button there. (Note that you can always get everything back from the "WP-Table Reloaded Dump File" that you created as a backup in step 1, in case you ever change your mind.)
7. If you are not satisfied, just deactivate TablePress again, and you will get the exact WP-Table Reloaded behavior as before. That should hopefully not happen though :-) And if it does, please let me know why, and I will happily try to fix any problems or improve the plugin.
About the arrow problem: TablePress uses a slightly changed approach here. Due to that, arrows will never overlap the text of the header cell, so that repositioning them is not necessary as the center position will always work.