When I am creating a new site, I have the option to create a new admin password, yet anything I put in there seems to have no effect... instead, it takes the password from the template site.

Is this a bug?

When creating a slave site from the front-end, the password is modified for the user login that is displayed in the front-end.
The user is the one that you have specified when you have defined the "website template".
When no user is specified, JMS select the first user that have the Super Admin level.(Normally the first user created by Joomla)

That's what I would have expected to happen, but that is not what happens.

I create a new site.

I enter a site ID: 9627436
Status: Confirmed
Owners: left blank
List of Domains: I put in the domains for the site
DB Prefix: 'template 1'
New Site Title: "New Test Site"
New Admin E-Mail: iputa@newemail.com (this doesn't seem to stick either)
New Admin Password: testing123

The end result is the site is created with the template1 admin password, rather than the new password I put in.

The only point that you don't have give in your sample is WHICH TEMPLATE have you used.

You just give a DB Prefix but which is the value of the template website.

You have to go in the "TEMPLATE SELECTED" and define which user will be used for the admin login.
If you don't specify it, JMS will use the first Super Admin of the website that is reference in the TEMPLATE SELECTED. Therefore the first Super Admin in the new DB prefix created.

You have several element to take in account
- The website that must by replicated (ie. originator)
- The website template that define the replication rule (ie rule)
- The new website to create "slave".

You define in the "rule" which "originator" is used and which login user of the "originator" must be changed during the new creation of the new "slave".
When you define a slave, you select the "rule" in the website template replication rule must be used.
If you don't specify the rule, this will be the first Super Admin that will be selected by JMS (the admin user created by Joomla)

So you are saying that I need to select an admin user in the TEMPLATE or in the SITE when I create the site?

Wouldn't selecting an admin user in the template cause every site generated from that template to have the same admin user?

As you replicate a website based on the originator, this will become the same as after the copy the user table are identical.
The "originator" is used because when you define the rule, the new website does not exists yet.