Is this correct: Non-kohanim may touch non-kosher animals (live or dead), including שרצים, but must avoid doing so when going to the Beis Hamikdash (for the holidays or otherwise).
Are kohanim always required to especially refrain from touching non-kosher animals (dead or alive) if they are not serving in the temple?
Is there any difference between kosher and non-kosher animals in these laws with regards to kohanim or Non-kohanim?
Today no one's doing Temple stuff, so it's a moot point.
Live animals are never "ritually impure."
Dead kosher species are ritually impure if they die of anything other than kosher slaughter; dead non-kosher species are impure no matter what. (Okay, at least something as big as a dog or cat.)
If someone's not planning on being around the Temple, they can be as tamei as they like, though back then the terumah taken from produce and given to the kohanim had to be eaten in ritual purity. So the typical kohen was expected to be tahor a lot (see first mishna in the entire Talmud, Brachos 1:1) ... but a given kohen was only on shift in the Temple two weeks a year.
The only "thou shalt not become tamei", if you're staying away from anything Temple or tithe-related, is for a kohen to be in close contact with a corpse.
Rashi Vayikra 11-8: יכול יהו ישראל מוזהרים על מגע נבלה, תלמוד לומר אמור אל הכהנים וגו' (ויקרא כא א), כהנים מוזהרין ואין ישראל מוזהרין. קל וחומר מעתה ומה טומאת מת חמורה, לא הזהיר בה אלא כהנים, טומאת נבלה קלה לא כל שכן. ומה תלמוד לומר לא תגעו, ברגל. זהו שאמרו חייב אדם לטהר עצמו ברגל One might think that Israelites are prohibited to touch a carcass. Scripture, however, says, “Say to the kohanim …[(a kohen) shall not defile himself for a (dead) person among his people]” (Lev. 21:1); thus, kohanim are prohibited [from defiling themselves by human corpses], but ordinary Israelites are not prohibited. Now a kal vachomer can be made: Since in the more stringent case of defilement by a human corpse, only kohanim are prohibited, then in the more lenient case of defilement by animal carcasses, how much more so [should only kohanim be prohibited! If so,] what does Scripture mean by, “you shall not touch their carcasses”? [It means that Israelites may not touch animal carcasses] on the Festivals [since at those times they deal with holy sacrifices and enter the Temple]. This is what [the Sages] said: A person is obligated to cleanse himself on Festivals. - [R.H. 16b, Torath Kohanim 11:74] (chabad.org)
It sounds like kohanim are forbidden to become tamei even from animals although I didn't see any of the meforashim speak this out and the normative halacha is not so.וצ"ע בלשון רש"י
Also see Hakesav Vehakkabalah there that says part of " קדושים תהיו" is to refrain from becoming tamei, even for yisraelim. (but perhaps once someone is tamei already this wouldn't apply) https://www.sefaria.org/Leviticus.11.8?lang=bi&with=HaKtav%20VeHaKabalah&lang2=en
