First, your demo reel should not be very long. I hire folks based on the first 20 seconds I see and I forget about them if they have not grabbed me by 30 seconds. Whatever you do don't throw in every trick in the book and don't use templates or repeat canned effects. Anyone doing the hiring has seen them all.
Second, Your sample reel should include your best work and start off with exactly the kind of stuff you want to be known for. Just throwing together a couple of "fan videos" is the fastest way I know of for your reel to end up in the waste basket. Anywhere you have worked should allow you to use work that you have done if you ask them as long as it does not violate the contract with the client. You should never pirate music for your reel. If you must use a popular tune don't do the whole song and give plenty of credit in the graphics. Personally I would never do that. There are hundreds of sites with royalty free and completely free music beds that you can use. I hope this helps. I started looking at sample reels and hiring people to make films in 1970 and nothing much has changed. If you can't grab them in less than 20 seconds you wasted their time.
If you are putting up a demo site to point perspective new clients to as a freelancer or production company then the same rules apply. Don't waste their time.
Thing is I don't know if this is a demo reel exactly. I'm planning on staying with a very niche field for the moment which requires I show an ability to visually accompany story driven narration, especially educational (history, science, literature, ect). And I wanted to have a number of examples available on a "portfolio website" that they can choose from. Insofar as my current contract I pretty sure it does say I cannot use the work I've done for them for "any reason" and honestly I'm not even sure I want anyone to know I've worked for them (they're that bad IMO). I was thinking about putting together a string of 2 or 3 second "snippets" of some of my most kinetic moments anyway on the assumption that if it's less than 4 seconds it's not a violation (I'm just making that up ). I'm not too worried about my current employer suing me for using my own work I did for them, I'm more worried about a potential new employer finding out I wasn't supposed to use it and not hiring me for ethical reasons. Would YOU not hire me if you found out I did what I described above? Using 2 or 3 second snips?
I would suggest getting permission from them once you cut up a good reel. They'll be able to see that you're not giving away their house. That way there is no issue of conscience on your part.
And, yes, you definitely want a demo reel of explainer video stuff. Just to demonstrate you have animation skills and to grab attention.
You know what could be fun? Make an explainer video about yourself and what you will bring to whoever hires you. "Hi, I'm whatever, and I do all these things. See that and that? Also, I can help you do this and that..."