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If this is an older GA account, before placing the Universal code onto your site you should check in your Property Settings -> Tracking Code to make sure that it is set to use Universal Analytics, if not you should start the upgrade process and wait until it is complete to replace any old GA code with the new.

If you have just created this account then yes, use Universal Analytics without worry.

You do not need to use any special code like that, all you need to do is make sure that the basic Google Analytics code is located on all pages on your site.  The one thing I do recommend doing is updating to use Google's newest tracking code, which should look like this:

You could create a segment for just traffic from that campaign (based on your utm_campaign) and make sure that generating the lead is set up as a goal. From there you can go to the Conversion -> Goals -> Goal Urls and see which of your product pages have conversions.

Segment Building:  https://support.google.com/analytics/answer/3124493?hl=en

Goal Set up:  https://support.google.com/analytics/answer/1032415?hl=en