From birth through citizenship application, I had several name changes; 2 through marriage and 1 via a legal name change. All had to be documented with official marriage/divorce certificates, and name change certificate. I don't know what type of name change you had, but I think DIA is going to want official documentation, not a newspaper clipping. You may do well to phone them and ask about your specific situation.
Regarding time spent in NZ, if the presence calculator says you're eligible, you should be fine. I applied for citizenship when I had my resident visa for 4 years and 240 days.
DIA counts years backwards from when you apply, rather than forward from the date your resident visa was first granted. (INZ counts forward) So, unlike going from Resident to Permanent Resident, where you had to wait to apply until after the 2 year mark was met, you can apply for citizenship as soon as you hit the time in NZ requirements.
For example if you (like me) were granted your resident visa and then never left NZ the entire time after, you would be eligible to apply for citizenship at 4 years and 240 days. That is the soonest point at which, counting backwards 5 years from the date of application, you could possibly meet the criteria.
To calculate whether or not you meet the presence requirement, you must therefore work backwards from the date that you intend to apply. For each of the preceding 5 one-year periods, you must ensure that you have been present in New Zealand for a minimum of 240 days as the holder of a residence class visa, and for at least 1,350 such days across all 5 periods. If you do not meet all of this, you do not meet the presence requirement on the given date, and must apply on a different date.
So for example, say you wanted to apply today.
*Did you have 240 days in NZ on a resident visa for the period 3/12/22 to 2/11/21?
*Did you have 240 days in NZ on a resident visa for the period 3/12/21 to 2/12/20?
And so on, back 5 years. Some of the years will need to have greater that 240 days to reach the total of 1350 days.
But like I said, the presence calculator does all that for you. Anecdotal evidence seems to indicate the calculator is more likely to tell you that you are ineligible when you are rather than the reverse.