Can I release my software under the MIT license?
If you use the GPL software in your software (combine two jobs) and then distribute a new job, you must distribute it under the GPL (copyleft).
There are different options for the GPL that have different effects on the functioning of copyleft (GPL, LGPL, AGPL), so if you need more information, you need to be more specific.
Should I include a GPL license (I'm sure the answer is yes)?
Yes. Any of the GPLs requires this.
Should I also include README with indications such as "part of the X file is licensed under the GPL, as it is an external library, the rest is MIT"?
Yes, this is helpful. However, if this is really a GPL, keep in mind that next to what you write correctly, the entire package should be distributed under the GPL.
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