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.
