Ok to independently investigate questions of interest to previous advisor?
Go ahead and work on the project on your own.  The fact that your former advisor introduced you to the general topic does not give him veto authority over where you will go with the topic in the future.

Having a former advisor who is interested in the problems should be an asset, not a liability.  If you find an interesting result, you have somebody more knowledgeable and senior in the field who you can show it to.