You will need to download the "current" version in the template downloads. Your 1.5 template(s) won't work in Joomla 1.6+. We have 2.5 versions of both the template and the shapeshifter available for Sportopolis.
As for Joomla versions, 1.7 and 2.5 aren't actually as different as the version numbers suggest. 2.5 is, generally fully backwardly compatible to 1.6. In short, if it works in 1.7, it will work in 2.5, and vice versa. It's usually better to keep up to date, as older versions will stop being supported sooner rather than later, but it's a personal choice.
It's the same with the extensions. The 1.6+ version is different to the 1.5 version, and will need to be reinstalled. This brings, in many cases, new functionality.
Now, having said all that you can, actually, update right from within Joomla. There is a guide on joomla.org -http://docs.joomla.org/Migrating_from_Joomla_1.5_to_Joomla_2.5
The caveat is that, miss one step, or do it wrong, and recovery is difficult. Just google it, or even search these forums for tons of posts from those who have had migration problems from 1.5 to 2.5. I'm not saying don't do it, I'm just saying it's as much a leap of faith as technology. I've personally done this on several sites, and it hasn't always worked first time. Database errors are common. They are recoverable, but it's still a pain.
Depending on your content volumes, you might consider installing the siteshaper, and then importing your 1.5 content afterwards. There are extensions on extensions.joomla.org which will do that for you. I can't vouch for it, as I haven't used it, but look at the J2XML extension. It seems well regarded, and may do all you need it to after a clean installation. It's free, too.
There are always problems with migrating across major versions. The idea is to minimise them. Unfortunately, with Joomla, it's easier said than done.
And yes, Modern Flavor will give you full responsive layout capability, regardless of the route used to update the site, or the version used. The template isn't available for 1.5, of course.