Machines generally run faster when they have a bit more free space than you have. The rule of thumb is to have 10% of free space, though that's a very imprecise rule, as 10% of a 128GB drive (12.8 GB) is just a bit more than 1% of a 1TB drive. You certainly don't need 60 GB of free space on your 600 GB partition to avoid performance problems. You might aim to have 15-20 GB of free space. That'll leave enough space to upgrade to MacOS Sierra when the time comes.
Note that, if you delete files from your HDD and depend on your Time Machine backup if you need them again... you don't have a real backup of those files. To be safe, move the files to a second external drive - the "backup" will still be in Time Machine.
You may also want to include that second external drive in your new Time Machine backups, especially if you're adding/modifying the files on that second external drive. Go into Preferences > Time Machine >Options... and remove the external drive from the Exclude list.
Your machine could be updated to Sierra, provided you clear enough free space. With Sierra, you can take advantage of Optimized Storage. It may save you space without buying extra storage in iCloud... but if you wanted to buy iCloud storage you'd have yet another option for managing space while waiting for the next MacBook model.