Eric, you can read/send email just fine in Europe with or without a SIM (or with any SIM) - that has nothing to do with email. You will get a different, Italian phone number if you get an Italian SIM; you'll get your Verizon number back when you put your Verizon SIM back into the phone.
Pease take a look at this long thread about using Google Voice and Google Hangouts:
Google Hangouts lets you make free calls to US phone numbers, even to landlines, while on WiFi or with mobile data. (Only works with a smart phone, though, not an old flip phone.) You can call non-US numbers took but it costs a few cents per minute. If you get a Google Voice phone number (free), you can also receive calls for free with Hangouts plus get Google voicemail, too. You could forward your Verizon incoming calls to this Google number while you are traveling, so you won't miss any calls and you can get your voicemail from Google. Or...just tell people your Google number to contact you in Europe (texts too).
You can install Hangouts (also Hangouts Dialer for Android phones) right now and use it in the US to try it out and make calls, etc. Works the same in Europe, except if you get a European SIM, you may need to add a +1 in front of US numbers to call them.
I already use my Google Voice number as my primary number anyway in the US, so when I go to Europe I really don't need to do anything except make sure I have a SIM (if need be) so I can use the phone without WiFi.
You might consider switching to Sprint - they have a special BYOD (Bring Your Own Device) offer right now to get a year of free unlimited service if you switch with a qualified device you own (so if your Verizon phone is paid off). Sprint gives you free unlimited roaming data (2G speeds) and 20 cents/min phone calls overseas. I just switched a month ago or so - pretty sweet deal. Costs me $3.20/month just for the taxes and fees.