A rollover from one IRA account to another does not count as a contribution.
A monthly contribution counts as a contribution whenever it is made, 2019 contributions may be made up to July 15 2020.
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You can’t contribute money into any kind of 401K except through payroll deductions. Technically, you execute a salary reduction agreement with your employer and the employer makes the contributions for you. 401K contributions are recorded when they are actually made, and you do not have the ability to make a 401(k) contribution in 2020 that will count toward the 2019 tax year.
An IRA is a completely different kind of account, a Roth 401(k) is not the same as a Roth IRA; they have different contribution and withdrawal rules. If you establish a Roth IRA, you can both; transfer your balance from a Roth 401(k) into the IRA account; AND ALSO make individual contributions into the Roth IRA, if you are eligible. If you are eligible to make contributions into your Roth IRA, then you can make contributions until July 15, 2020 and designate them for the 2019 tax year, this may require extra attention when you make the contributions with the trustee to make sure they are credited to the proper year. Then, you can also make 2020 contributions up to the eligibility limit.
The ability to do a rollover or transfer is separate from your ability to make contributions.
You should do a direct institution to institution transfer, rather than a rollover, if at all possible. There is less risk to you.