If the check is made out to your new retirement account, say to an IRA at a particular financial institution for your benefit, even though the check is given to you for forwarding to the financial institution, it constitutes a direct rollover and would not be subject to mandatory withholding. It would also not be subject to the 60-day rollover deadline, but there is no good reason to delay forwarding the check. If it's a direct rollover, it needs to be deposited to the type of account that you specified when you requested that the 401(k) plan make the direct rollover. Depositing the distribution to a Roth IRA when the request was for a direct rollover to a traditional IRA would be improper if the check is made payable to a traditional IRA (but it does happen from time to time and complicates reporting).
If the check was made payable to you personally, you do have the concerns about substituting other funds to complete the rollover of the portion withheld for taxes and the 60-day deadline.
A distribution from a Roth IRA that comes from a taxable conversion (a conversion that adds to AGI) *is* subject to a 10% recapture penalty unless held for 5 years or you reach at 59 1/2.
