You may notice the black residue if you fry eggs in the pan as well.
Most likely, the black residue is charred (greasy) food sticking to the seasoned oil. Since Flaxseed oil has low heat tolerance, it could be that disintegrating, too. Otherwise it could be related to the iron in the cast iron which isn't bad for you (some say even good).
a) Is it normal: Yes if you keep with the same regiment (recommend you don't).
b) What is Wrong: Water won't perfectly wash non-polar chemicals such as burnt solids in grease (milk solids while making ghee), so the hot water no-soap routine leaves the stuff in pan.
c) How can I fix it: Try the following going forward:
After you finished cooking and while the pan is somewhat hot, put a heap of salt in the middle and with a paper-towel spread and lightly scrub the pan. The salt will pick up the would-be black stuff, largely disinfect the pan, and the abrasive properties of the salt helps with the cleaning.
Wipe off the now brown salt from the pan and rub-in the coat of oil as you normally do. The new coat will stick better to through the salt's abrasive effect.
This routine won't eliminate residue altogether. But it seems to work great long term.
Note on oils:
Flaxseed oil is best consumed cold and within three weeks. I haven't heard of a credible source promoting seasoning the pan with flaxseed oil as it has one of the lowest smoke points in the oils. Use a more heat tolerant oil like light (not extra-virgin) olive oil or canola.
Lodge claims their pans are pre-seasoned. It's a great company, but don't believe that pre-seasoned malarcky. Yes, they do treat the surface with oil, but not nearly enough. No worries though, you really only need a good seasoning layer on the bottom, cooking surface.
Scour the cooking surface of the pan thoroughly, then add a film of oil to the bottom of the pan and heat it on a burner. Move it around frequently to minimize hot spots. Let it get to smoking, allow to cool enough to touch, wipe it out, and then a add a bit more oil and smear it with a paper towel. Repeat 3 or 4 more times. Ignore the walls and handle of the pan for now, just thoroughly re-season the whole thing when you have access to an oven.
Opinions vary as to the best oil to use. My personal favorite is lard, flax seed is often highly recommended.
Read this question and answers for more info: What's the best way to season a cast iron skillet?
Congrats on your new pan. Cast iron is lovely, I have a set of skillets that belonged to my grandmother, my mother and now me. Cast iron takes some "getting to know" but once you know your pan you'll love it. BTW, except for the first time scouring, ideally your pan should never see soap or detergent. There is a lot of good info here about cast iron, just use the search engine to learn more.
EDIT: One more thing. All new cast iron pans are treated in some way before they are sold to prevent rust. In the case of Lodge, that treatment is a light seasoning with food grade oil. There is nothing that needs to be removed before first use. In fact the seasoning step I recommend here in not technically necessary, it just gets you to that black, old, almost-as-slick-as-teflon surface faster than repeated use alone. I only recommend scouring the cooking surface before you season the first time to give the factory surface a tiny bit of "roughening up", giving your first seasoning layer a bit of something to grip on to. So don't scour the whole pan, just the cooking surface. By the time you have access to an oven, the factory surface on the rest of the pan will already have developed some patina just from use, so you don't need to scour to season the rest of the pan, just oil the whole pan.
Some cast iron pans from other companies are not preseasoned, they're coated with wax or some other thing you don't want in your food See: Do all new cast iron pans and skillets have a protective coating on them, which must be removed? Those pans have to be scoured all over before first use. Ideally, then they would be seasoned in an oven before first use. If you had a pan like that, but no access to an oven, my advice to you would be the same accept that you would need to scour the whole pan, season the cooking surface and lightly oil the whole pan every time you heat it for the first several times, periodically thereafter.
Best Answer
In the restaurant, we just poured some coarse salt into the pan and put it on high heat (gas range). Then tossed the salt around and poured out the results.
At home, with the electric range top, I put my carbon steel wok on a burner on high (it's scary, but it works) and watch as everything burns off and the carbon steel "steel" look returns.
The brilliance doesn't last but the surface does. I use a cotton rag that is very lightly dabbed in peanut oil to brush the surface while it's hot.
And then it darkens when it's cooling/cooled.
Don't hesitate to go hot, really hot. Just be safe at the same time.