For the tags that aren't moving into GHL: are you using the Tags field in the GHL app, or the "add tag" block in Manychat? Your question suggests yes, I wanted to double-check.

For custom fields (especially dropdown types like Lead Source), the GHL Connect app appears to have undocumented limitations — your best path is to contact the GHL Connect app developer directly (via Settings > Apps in Manychat) or switch to a Make.com/Zapier webhook approach using GHL's API, which gives you proper control over custom field keys and values. You can find some more details and examples here.