Yes, this is perfectly normal.
Given your gravity is 1.003, I'm not sure if fermentation actually started, or you just caused some of the CO2 to come out of solution - in both cases you'll get bubbles in the airlock. Ah, 4 days you say, that's quite a long time. It could be that you got "stratification" - bands of higher concentrated sugars at the bottom. By rousing the yeast you give them more contact with the sugar. You'll know for sure if it really was still fermenting by taking another reading in a couple of days after airlock activity has subsided.
The "gunk" on the bucket above the wort level shouldn't really be mixed back into the wort. It won't harm you, but it may taste bitter. I thoroughly recommend John Palmer's book, How to Brew, and IMHO should be considered required reading before attempting to brew. In it, he mentions about the crud:
The brown scum that forms during fermentation and clings to the side
of the fermentor is intensely bitter and if it is stirred back into
the beer it will cause very astringent tastes. The scum should be
removed from the beer, either by letting it cling undisturbed to the
sides of an oversize fermentor, or by skimming it off the krausen, or
blowing off the krausen itself from a 5 gallon carboy. I have never
had any problems by simply letting it cling to the sides of the
Please post back if this affected your cider, but keep in mind that the cider itself may be a little bitter, not just because you stirred the trub back into the beer. 1.003 (or lower if it fermented on) is pretty low - depending upon your original gravity - there may not be much sugar left to provide sweetness.