WordPress is licensed under GPLv2 according to their website. Answers to your questions, under the GPL:
1) Am I OK to publish my version of the code on GitHub/Packagist?
Yes, as long as you keep a copy of the GPLv2 license with your released code.
2) If so, I presume I need to use the same license? For instance, I cannot change the license to MIT, which is what I would like to do?
This I’m unsure of, and the GNU GPLv2 FAQ isn’t incredibly clear on the topic, at least in terms of changing the license. My best guess is that you would have to remain under the GPLv2 license, with no ability to change to a different license.
3) Is my client’s code (internal use and public facing only; nothing developed for resale) affected in any way by using this component? I’m pretty much using commercial, self-developed or MIT licensed components throughout.
Again, I’d refer to the FAQ as it might be more helpful than I am. If you’re using GPL code and want to distribute any part of your code, you must include the source and a copy of the GPL. As far as usage, I don’t know that you need to make any changes to how you are doing things now.
You may be able to get some answers from these GNU-related IRC channels, or find a lawyer who specializes in free software licensing. You can also probably find forums that provide this information or discussion about it for free.
Legal disclaimer to the above advice: I am not a lawyer