Of course. The heart rate is directly related to sickness and health. It is affected by fever--the normal accompaniment to many illnesses--and by stresses of all kinds.
When we are sick, the functioning of most of the organs of our body--lungs, kidneys, gut, for example--are affected. The heart is an organ. It is affected by all those changes, too.
It's normal.
Yes. If I have the flu I will lie there with ectopics going on a lot of the time. My current spate of ectopics this last week I'm wondering is due to some milder infection.