1. Maybe and Yes. If the only thing that changes is form 8606, you can file that alone (it has a signature page). If other things change, you must amend the entire return (such as, the taxable income from the bonus you describe).
2. For prior years not online, you can purchase Turbotax to install on your own computer, from the link below for "products for prior years." If you already paid to file online for that year, try calling support and see if they will push a free copy to your account. If you need to go farther back than Turbotax supports, you will need to amend manually or see a professional. It's not too difficult to do manually, just make sure you download the correct forms and instructions for each year from the IRS web site, follow your prior return, and make the changes needed. With an amended return, you don't need to submit everything all over again, just the forms that change (1040, 8606) and the 1040-X cover page that summarizes the changes.
If you make a contribution for 2021 in March 2022, that is reported on your 2021 tax return (form 8606). Then, if you make another contribution in April 2022 for 2022, and convert the entire amount, the 2022 contribution and the conversion are reported on your 2022 return. Contributions are always reported for the tax year you made them (including if you made them retroactive); conversions are always reported exactly when they happen (because conversions are never retroactive).
3. If you make a $6000 non-deductible contribution to a traditional IRA, and convert more when you do the conversion (because of a promo, or investment gain, or any other reason), that extra amount is taxable income. It's calculated on form 8606 and transferred to form 1040 line 4b (taxable amount).
