1. It may be helpful to look at the fine detail of the instructions about character for the temporary and the residence visas.
http://www.immigration.govt.nz/opsmanual/35052.htm
What is normally done is for the applicant to submit their application along with the police check certificate which shows the offence(s). Then the INZ official will contact the applicant asking for their comments on their record. They then have to come up with a reply which gives an account of their circumstances at the time (so, any extenuation possible), their present-day attitude to what they did, and how they have now changed so such a thing would never happen again. This is when they would also submit any character references they can obtain, preferably from people of standing in the community who have known them for some time (that is, not just family or friends).
2. I don't know how long a partner-sponsored visitor's visa gives you to arrive.
You can apply for Residence at any time when you can show evidence of having been living together for 12 months or more, so you could do it now or at any time you want to, regardless of whether you have a temporary visa first.