If you extend, yes, you have up until the extension to recharacterize your Roth contributions. Note that this is the last year that this is in effect because the recent tax act removed the ability to recharacterize your contributions to a Roth in the following year.  

See this info at the IRS here: https://www.irs.gov/retirement-plans/ira-faqs-recharacterization-of-ira-contributions

Can you file a 2017 Federal extension late if you did not owe a balance? I may need to recharacterize my 2017 Roth conversion-can I do that by October 15 2018 even without filing the Federal extension?

"The election to recharacterize and the transfer must both take place on or before the due date (including extensions) for filing your tax return for the tax year for which the contribution was made to the first IRA."
You missed the deadline to request an extension.

Did you convert IRA funds to a Roth? or did you make a contribution to a Roth?

If you neither filed your tax return nor requested a filing extension by the regular filing deadline, you are not now permitted to recharacterize because you did not meet the requirements for the recharacterization deadline to be extended beyond the regular filing deadline.