Yes to both. PSP Go (PSP-N1000) has:
| PSP-1000/2000/3000 | PSP Go | |
|---|---|---|
| Battery | Removable (1200/1200/1200 mAh) | Internal (1200 mAh) |
| Charger | AC adapter (barrel) | AC adapter (barrel, same) |
| USB Charging | No | No |
Keep AC adapter — replacements are model-specific and scarce.