Angie - I brought my dog with me last fall to Italy and can tell you what I experienced.
I checked with numerous airlines because even though I might have booked with one airline - odds were I would be taking at least one leg of the flight with a codeshare airline.
First, in order to take the Lira along with you in the cabin she must be able to fit into one of the approved
pet carriers which can fit under the seat in front of you. (Although the size is approved, I have to say that on a couple of planes it was a bit of a squeeze to get the carrier underneath)
The maximum weight I could find on any of the airlines was 15 lbs including the carrier. KLM had the lowest weight limit at 9 lbs. I will say, however, that no one weighed the dog or checked the size of the bag or whether or not she could stand up inside it.
I finally booked my flight on United with Lufthansa codeshare and was told all was OK with the dog. About a week before departure I called just to verify I was bringing a pet with me and was told that I was not approved for the Lufthansa legs of the flight. United promptly switched my flights around for me to accomodate.
On my first leg of the flight I was fortunate to be placed next to an empty seat so, once we took off, I put my dogs's carrier on the seat next to me and she was quite comfortable. After that - I asked when I boarded if I could be placed next to an empty seat in order to keep the dog from crying and disturbing other passengers. It worked and we were both happier. I was able to unzip the top of the carrier and she could stretch out or lay down as she chose.
One kind stewardess even told me where to find the handicapped lavs so I could take her in there and maybe have a little room for her to walk around a bit and reliever herself (I brought along the little pee-pee pads).
I don't know how much Lira weighs but if she is small enough I would opt to bring her in the cabin with you. Each of the airlines websites have a section devoted to travel with a pet. You don't need to muzzle to animal. In fact, I have seen on a couple of the sites where it is not allowed at all to muzzle them.
I believe if they have to ride in the cargo hold there is also a special carrier required and that is listed on the airline websites too.
Once we landed at each destination I went thru customs the same as any other time -- no one asked for anything special. ((I did have to wait in a special line for about 30 minutes upon returning in Chicago but all for nothing as I was allowed to go right thru once the security agent actually showed up))
When checking in for my departure flight I showed her health certificate but that was it. Going thru security I had to take her out of the carrier and let it move thru the x-ray machine and carrier the dog thru with me. Then wait the obligatory couple of minutes while everyone says "Hello" to the dog.

All-in-all it was a very pleasant and easy experience.
Tori