How do I remove an alias from my Apple ID?

From your question I assume you already have an @me.com address (which is automatically an Apple ID) - is this correct? When iCloud was introduced @me.com (and @mac.com for those who have them) accounts had @icloud.com addresses added to the account (all having the same text before the '@'). All three are effectively the same address and you cannot delete one of them.

If the problem is that, while you are using the @me.com address, you are getting spam to the @icloud.com address, you can handle this by setting up a Rule on the iCloud website to move all incoming messages addressed to the @icloud.com address to the Trash, from where it will be automatically deleted after a week. (Go to the Mail page at icloud.com, click the cogwheel icon at bottom left, and choose 'Rules').

Your @icloud.com address was created when you created the iCloud account. You can't change or remove it. You can get a different address by creating an email alias as I indicated above and using that, and just ignoring the existing iCloud address.

I have a similar problem. I don't ever use the icloud mail, but a few years ago noticed that somehow an apple id alias (ending in icloud.com) was attached to my account (I always use a non-apple e-mail as my primary). I have no idea how it happened - but at one time I think I was hacked and had to change my password. As such, I have only one icloud.com e-mail with the account and it is someone else's name who I've never met. I have to remember to change Mail whenever I set up a new device so that it does not use that by default. It hasn't been a huge problem, but a bit irritating. I was hoping that there might be a fairly easy way to remove that but I guess not.

You can only have an @icloud.com address if at some stage you opened an iCloud account - this would have been on your iPhone as you can't do it on a PC. iPhones encourage you to use iCloud for backup and syncing purposes. You can sign out of iCloud on the phone in Settings>iCloud (it may say 'Delete account') but of course you will lose any of the associated facilities.

Your Apple ID has the movies associated with it from being signed into iTunes. You can change the iCloud account to using a different Apple ID without affecting iTunes, though most people prefer to use the same ID for both.

If you sign out of the iCloud account the data which will be removed from your Mac will be any synced contacts, calendars, photos, iPhone backups or emails. If you aren't using iCloud for any of this all you need to do is to go to Settings (or System Preferences)>iCloud and sign out there.

iCloud is used to sync the data I listed between devices, such as between a Mac and an iPhone, and provides an email address. Many people find it convenient, but there is no obligation on you to use it if you don't want it.

iTunes and iCloud accounts are quite separate. They can have different Apple IDs or they can have the same, it's entirely up to you. So changing the iCloud account won't affect your purchased movies or songs from iTunes. You will have to download any pictures which are on iCloud back down to your Mac, and the same for any synced data such as contacts if you enabled them; then you can sign out of iCloud. You will need to create a new Apple ID, for which you will need a different non-Apple address. Then you can sign in with that in the iCloud Settings pane and create a new email address. Then you can re-upload your pictures and, if so desired, sync your data. You will now have different logins for iTunes and iCloud.

Does this work if I have a PC? I have not had a Mac in years, since my Mac mini's disk died and Apple refuses to come out with a new one.

I would have thought so but as I don't use this facility I can't pontificate on it - you could make this a separate question.

If the iTunes and iCloud accounts are separate, why can't I just associate the new iCloud account mynewname@icloud.com with the old iTunes account? It seems really bizarre that I would have to have two different Apple IDs.

That @icloud.com address is inextricably linked to the Apple ID which created the iCloud account, so you can't make it into a different Apple ID. You will have to create a new email address - a Yahoo or GMail one would do - and use that to create the new ID and then iCloud account.

Also, you have said that many people choose to have the same email addresses for iTunes and iCloud. How is this possible if iCloud has to be an icloud.com address and the Apple ID has to be a non-Apple address?

The iCloud account is created using an Apple ID which must be a non-Apple address. The same, or a different address, can be used as an ID for iTunes. The @icloud.com address is the email address to be used with the iCloud account, not the ID itself.

You would find the whole process a lot less complex if you simply created an email alias as I indicated earlier, and then used that as the email address. You would simply ignore the original@icloud.com address from now on. (It will still exist if you create a new iCloud account, so it comes to much the same thing - you can't actually delete it altogether.)