Does Ivern Friend to All get the extra point if I have units with all the different tags at the same battlefield as him, or can they be anywhere in play (e.g., bird, poro, and cat with Ivern at battlefield and dog at base)?

Yes — they can be anywhere in play under your control.

Ivern's ability says "if your units have all of the following tags among them," not "at this battlefield." So as long as you control at least one Bird, one Cat, one Dog, and one Poro across your base and all battlefields, the condition is met (FAQ #10315). A Dog at your base counts just the same as one sitting next to Ivern.