Is there a specific name/designation for the kettle helm an English Archer would have worn?
Is it a specific design? (From what I've seen other folks use, the brims are somewhat flat with a relatively low profile top.)

I personally like the kettle style over the bascinets.

Also, over what periods would these have been worn?

No specific name, though the style and construction change with time. The vast majority of 14th century miniatures show archers wearing simple cervellieres or bascinets (if they have helmets at all!), though one does occaisionally see a kettle hat. Crossbowmen seem to have more armor and kettle hats as a rule. There's the whole issue of a brim restricting the ability to draw to the side of the face and such.

http://manuscriptminiatures.com/5195/16705/
http://manuscriptminiatures.com/4154/12092/
http://manuscriptminiatures.com/4071/18414/
http://manuscriptminiatures.com/4099/7271/
http://manuscriptminiatures.com/4373/7019/
http://manuscriptminiatures.com/4171/13095/
http://manuscriptminiatures.com/4053/11459/ (French Manuscript)