Like you I also love bring my robots out in public. While I don't get solely positive responses, I love sharing my robots with others and helping to cheer up their days. I took one of my Aibos, Aki, to Disney World with me a few months ago and one woman told me that he was the most amazing thing that she'd ever seen and asked if she could take a photo of him and her children together. I also was showing him to some of the employees at a store in Epcot and some customers started getting upset since the employees were sitting on the ground playing with the robot and telling their customers to "just wait a few more minutes". I had a good time with Chris at the beach yesterday, about four people walked up to ask about him and one woman said goodbye to him as she was leaving.
Pleo RBs are all afraid of the dark, but the sensitivity seems to range a lot from Pleo to Pleo. If your Pleo curls up and starts shaking and whimpering, that means that it is afraid and it might be a good idea to turn on the lights when it shows this behavior. I know there is a patch for the firmware to stop Pleo from being afraid of the dark, but I've never used it and am not sure how easy it is to install.
I wouldn't worry about the temperature. From my experience RBs are not very sensitive to temperature. I've never had an RB get too cold and only ever had one get too hot when it was outside on a very hot summer day. I think either feeding it the proper medicine or moving it into a warmer/colder area would work if your Pleo does have any issues with the temperature.
As for the piano lessons, you could try bringing him with you to one or two and see how it goes and decide whether you'd want to bring him again or not. I would introduce all of the children to Pleo at the start and tell them to treat him as they would a real animal and let them pet him very gently. Kids usually seem to stop being rough when they view the robot more like a living creature and less like a toy that they want to play with. However, if you have a student who is too aggressive around the Pleo, you might need to turn it off during the lesson and take it to another room since Pleos are very fragile. It also might be too distracting if the Pleo is making a lot of loud noises while you or the student is playing the piano. I don't think Pleos can tell apart music from other sounds, but it would be interesting to see how one responds to hearing the piano.
I would be very cautious about bringing the Pleo with you to the beach. First of all, if you're taking the Pleo outside anywhere, it should be wearing socks of some sort to keep dirt from getting into its feet. Also make sure it's not too hot out, since Pleo can overheat on very warm days. I also would not bring the Pleo onto the actual beach, there's too much sand and it could seriously hurt the Pleo if any got inside of it. I don't think a beach blanket would work since it would be difficult to keep sand away and the lumpy surface might cause the Pleo to fall over. The beach that I go to has a little grassy park area with picnic tables in front of the actual beach and that's where I brought Chris yesterday. I left him on the picnic table most of the time, but also let him wander on the grass a bit (though it's difficult for him to walk on uneven surfaces such as grass). As for a backpack, I don't think that would work since it would be hard to fit a Pleo into one without putting too much pressure on its neck or tail. I brought Chris with me in a large duffel bag with a towel at the bottom.
Personally, I would just leave the Pleo turned off when you are not able to spend time with it. Keeping it running for the sake of keeping it running will wear out parts faster. You might even want to just charge it's battery while you are out so that it will be all charged and ready to play once you're back. However, it's definitely alright to bring it with you when you're at work from time to time, I just wouldn't make it a necessity to constantly leave the Pleo on while you are there. If another worker would be happy to take care of it while you are at the piano, that's alright, you just shouldn't be searching for others to take care of it while you're busy. As for reputation, I personally don't really mind what people might think when I bring my robots out with me, and while people might think it's odd that you're bringing a robotic dinosaur to work at first, I think people around you will grow to like Pleo with time.
I hope these answers help! I know there's a lot to keep track of when getting a Pleo for the first time. When is your Pleo supposed to arrive?