Do "dread to do something" and "dread doing something" mean the same thing?
Not quite exactly the same, but similar.
When considering the gerund vs. infinitive, we usually follow 'dread' with a gerund, where it means fearful or reluctant to face a particular action. We usually only follow it with the infinitives 'to think' or 'to consider', where it means something more like "don't want to think about/consider X".
- He can't swim and dreads to go going in the water. Grammatical, but we'd normally use a gerund.
- I dread thinking to think about what they might do next. Again grammatical, but normally we'd use 'to think'.
How do these sentences sound?
They're not wrong, but I'd normally expect to see the suggested gerund/infinitive instead.