Unless you are absolutely still (like lying down) and thinking about little or nothing, your heart rate will vary... if you burp, fart, hiccup or twitch, it may show in your heart rate. Some thoughts that might make you feel slightly stressed or excited can result in a spike... as can sudden or loud or unexpected noises (someone talking to you, a car outside, a door slamming etc)
It will def fluctuate as you walk about.

I don't wear bra when cycling, I wear a crop top. And the heart rate monitor belt fits underneath that, I make sure the bottom elastic of the crop top sits just under the bottom edge of the belt and it ensures that the monitor stays in the right place.

My partner never wears it at nipple height, he wears it about half way between the bottom of his sternum and nipple height. I just place it under my breasts in the line where bra support would usually be.