Regarding #1, TurboTax never treats code H as taxable. A code-H Form 1099-R is required to have a zero taxable amount shown in box 2a of a code-H Form 1099-R.
Regarding #2, code G indicates a rollover from a traditional 401(k) account. Rolling this over to a a Roth IRA is taxable and this Form 1099-R should show $1,603 in box 2a. My guess is that #1 and #2 came from the same plan and #2 is reporting the rollover of the pre-tax employer matching contributions (which were only permitted to be made by the employer to your traditional account in the plan).
Regarding #3, code T means a distribution from 1 Roth IRA, not a distribution from a 401(k). If this truly came from a designated Roth account in a 401(k) plan, you need to get a corrected Form 1099-R from the plan.
Regarding #4, this is a distribution from a traditional account in a 401(k). No matter what you do this his distribution other than rolling it over to another traditional requirement account, the distribution is taxable. Rolling it over to a Roth IRA is taxable.
Note that these are all considered "income" but if all were rolled over to a Roth IRA, only the distributions form the traditional accounts are taxable income. To see which ones are being taxed, review Form 1040 lines 4b and 5b.
Thank you for taking the time on this. Your assistance has been appreciative.
Barbara