Can I run MacBook Pro Retina 13 inch early 2015 without a battery? I have a swollen battery, and want to remove it while waiting for the replacement. 

Can I remove it and safely turn on the Mac only with the power chord?
Most Apple laptops will work safely, but slightly less reliably, with the battery removed. They will behave like a desktop computer that possibly has an inadequate power supply for full load conditions: remove the power, and everything goes poof, which is not ideal, given the ease with which a MagSafe or USB-C connector can disconnect. Repeated 'shocks' of this kind to the circuitry may cause damage to the filesystem.  So remember to Shutdown before removing the cable and keep good backups. 

Some people say that the Mac may run at a lower clock speed, to reduce power load, though this may depend on model.

However, removal of the battery in this model is not easy. If you're planning on replacing it yourself, then crack on. Otherwise, you might just want to leave it alone until the repair is carried out.