Can many "regular" animals interact with magical people, creatures, and places in Harry Potter world?

Question

This question was inspired by another question on animals in the Harry Potter world. That question was about animals that were half-magical and half-muggle.

Which animals that look and act mugglish (is that a word?) can interact with magical people, beasts, and places?

I am specifically not asking about truly magical creatures that had interacted with muggles incidentally. So that leaves out Thunderbirds, phoenixes, occamies, hippogriffs, dragons, and other such beasts.

I am asking about animals like owls which can fly from a magical place like Hogwarts to a muggle place like London and back.

Can other muggle creatures go from the muggle world to the magical world and back?

Edit to clarify: I am not asking about situations where somebody brings an animal to a magical place, such as when Neville brings his toad, Trevor, to Hogwarts. I asking about where animals can freely interact with both magical and muggle places on their own. We know muggle humans can't, because otherwise they would see Hogwarts or enter Platform 9 3/4. But we don't see incidents in the stories where muggle animals try to go to magical places on their own and are prevented from doing so because of charms and spells.

Edit to clarify: Although an animagus like Peter Pettigrew can transform into a rat and go from magical places to muggle places, I am only asking about normal animals. It seemed obvious to me that animagi could go where they wish in either human or animal form. (Thanks to Reya for pointing that out.)

Answer 1

Yes; or at least you would think so

There has been no mention or reference that "muggle" animals are in any way prohibited or barred from interacting with the magical world, let alone that it would be inconceivably impractical to enforce such a limitation.

In fact, here's Newt Scamander talking about how difficult it is to keep the 30 or so magical species secret from the muggle world:

The International Confederation of Wizards argued the matter out at their famous summit meeting of 1692. No fewer than seven weeks of sometimes acrimonious discussion between wizards of all nationalities were devoted to the troublesome question of magical creatures. How many species would we be able to conceal from Muggle notice and which should they be? Where and how should we hide them? The debate raged on, some creatures oblivious to the fact that their destiny was being decided, others contributing to the debate.
At last agreement was reached. Twenty-seven species, ranging in size from dragons to Bundimuns, were to be hidden from Muggles so as to create the illusion that they had never existed outside the imagination. This number was increased over the following century, as wizards became more confident in their methods of concealment. In 1750, Clause 73 was inserted in the International Statute of Wizarding Secrecy, to which wizard ministries worldwide conform today:

Each wizarding governing body will be responsible for the concealment, care and control of all magical beasts, beings and spirits dwelling within its territory's borders. Should any such creature cause harm to, or draw the notice of, the Muggle community, that nation's wizarding governing body will be subject to discipline by the International Confederation of Wizards.

^{-Fantastic Beasts and Where to Find Them, Introduction by Newt Scamander (xiv - A BRIEF HISTORY OF MUGGLE AWARENESS OF FANTASTIC BEASTS).}

and continued:

It would be idle to deny that there have been' occasional breaches of Clause 73 since it was first put in place. Older British readers will remember the Ilfracombe Incident of 1932, when a rogue Welsh Green dragon swooped down upon a crowded beach full of sunbathing Muggles. Fatalities were mercifully prevented by the brave actions of a holidaying wizarding family (subsequently awarded Orders of Merlin, First Class}, when they immediately performed the largest batch of Memory Charms this century on the inhabitants of Ilfracombe, thus narrowly averting catastrophe.

^{-Fantastic Beasts and Where to Find Them, Introduction by Newt Scamander (xvi - MAGICAL BEASTS IN HIDING).}

Now imagine how difficult it would be to prevent animals going the other way, from the muggle world into the wizarding world, and vice versa.

I'm sure there are examples somewhere, but it's difficult to pick for the following reasons:

• Fully magical places - places which have become unplottable, hidden or just generally constricted to magical only - are usually too far from "muggle" locations for "normal" animals to venture in and out from.
• The books are written from the perspective of Harry Potter, unless he specifically notices or mentions these things, it's hard to get examples