mai is used very widely. taw is more common in some south and west Wales dialects.
No. This is explained in the notes for this section, 'Comparing3'.
mai/taw is a special particle used to introduce an emphatic clause following phrases such as 'I think/said/ believe/know that....'. - in that particular context you could think of it as the 'that'.
It is not a verb or verb-noun at all, and so nothing to do with mae. The clause that is introduced by mai/taw has to include its own verb.