You have a new chain, how new? Some chains will stretch badly in the first few minutes of running, I recently had an Oregon chain stretch so much in the first five minutes it jumped the rim sprocket. They can continue to do so in the first half hour of cutting, you may have to adjust the tension several times during this period.
As for the unevenness of tension as the chain moves around the bar, this is caused by runout of your rim sprocket, your drum, or a combination of both. I find the cheap chinese rims are bad for this, even the powermate ones have it sometimes as well.
If both, sometimes you can get them to cancel each other out by changing the position of one in relation to the other, trial and error by rotating the rim one notch at a time on the drum and trying it again. A royal PITA, but sometimes it works.
The rubber part on the tensioner is there to keep the screw from vibrating its way out of the saw. An o-ring will do the same job. Some adjuster setups use a locknut to keep friction on the screw to prevent it from turning under vibration.
Once your bar is tensioned and the bar nuts are snugged down the tensioner's job is done, and it normally doesn't matter if it comes loose as the bar should be held secure by clamping force. I have had this happen and the proper chain tension remained.
Proper tension is just enough to take the slack out of the chain. When you go to tighten the bar nuts, pull up hard on the top of the chain first, and hold it there as you tighten them. Check the average slack as you move the chain around the bar. Adjust again if necessary. It should move freely with out binding.