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If I purchase an OEM copy of Lightroom 6.0
You mean a perpetually licensed (standalone) copy? i.e. not a Cloud version.
The answer is yes. All updates within the same version number are free. 6.0 > 6.1, 6.2, 6.3, 6.4, 6.5.1 etc.
Lightroom 6 gets all the same bugfixes, new camera support and lens profile support as the Cloud version. Lightroom 6 does not get new features such as Dehaze and Boundary Warp.
Lightroom 7 (if there is one) will be a paid upgrade.
If I have the Cloud version, would Lightroom 7 be a paid upgrade?
No, once you subscribe for the Subscription whenever there is a release for any upgrade for your purchased product you can download straight away from the Creative Cloud Desktop App without paying any upgrade fee.