Sourdough starters are rarely completely ruined, unless you're growing significant amounts of mold or something.
It is possible that your refrigerated "break" early in establishing your starter ended up hurting the yeast population and accidentally selected for something else (perhaps undesirable bacteria) that is now growing and creating odd odors.
It's also possible that your starter is perfectly healthy and well-established. A strong starter should often rise and begin to fall in 12 hours or less. Mature starters tend to go through a cycle where they smell yeasty as they are expanding, have alcohol notes around the time of collapse, and then acidic (vinegar/acetic acid and lactic acid) notes as they age further, due to bacteria converting the yeast waste products into acids.
Have you tried to bake with your starter yet? I'd see how well it works and whether it can successfully leaven bread dough. If so, it's probably fine. Starters do often go through periods of odd smells and weird behavior in the first couple weeks, but they'll stabilize after more regular feedings as a more consistent set of microorganisms becomes permanently established.
You don't mention any liquid floating on top ("hooch") or any discoloration. If those were appearing, I might be a little more concerned. If not, try baking a small loaf with it, and see what happens. My guess is that it won't be very sour, but with a feeding regime like you have, it probably shouldn't be.
Lastly, you asked about whether more frequent feeding is required. I don't think so, given how much you dilute your starter during feeding. Twice per day feedings are recommended at room temperature for those who do something like a 1:1:1 (i.e., 1 part starter:1 part flour:1 part water by weight) or 1:2:2 feeding. You're doing something closer to 1:8:8, which means it will take longer for the yeast to process all the new flour and then for the bacteria to deal with those waste products. The only danger with such high dilution in young starters is that microorganisms actually present in the flour may still be able to overwhelm the things you want to grow and/or the starter may not be developing enough acidity by the end of your growth cycle to kill off those bad things.
Does your starter smell (or taste) acidic right before you feed it? (It doesn't have to be strongly acidic, but at least mildly so.) If so, I think your starter is likely very healthy and having youthful growth spurts.
In any case, I'd test it in a batch of dough before trying to diagnose further. In a worst case scenario, as long as you're getting some acidity before feeding (and not seeing mold, hooch, discoloration, or other weird things besides the odor), you should just keep feeding regularly, and it will likely sort itself out in a week or so.
Don't over think this.
If you know somebody who already makes sour dough, they need to refresh their culture once or twice a month. Get their "throw away" starter and use it to start yours. I bought a starter from King Arthur about 10 years ago. Still using the same starter today. King Arthur still sells sourdough starter. Their crocks work well too.
You can start a starter simply by putting out a bowl of water in the summer air. The yeast that's in the air will find its way into the water. You should then add some of that water to perhaps a half dozen jars of water flour mixture. Not every jar will start but the ones that start bubbling are worth keeping and nurturing. It may take a day or so at room temp before you see bubbles. After about two days the mixture will stop bubbling. That's when you need to add more flour/water mixture.
I use King Arthur unbleached bread flour in the light blue bag. Use 1 cup flour and one cup tap water and 1 cup starter. You can boil it if you want but I never do. Just make sure you don't add boiling water to the starter or you'll kill it. Tepid or lukewarm is fine. I don't worry about chlorine, add honey or lower the pH. If your water is unusually alkaline you may need to do something about it but first you would need to know if it's alkaline or acidic. Inexpensive pH test strips are available at pet stores that sell tropical fish, especially if they sell saltwater fish. Running your water through a carbon filter will remove the chlorine (Brita, PUR, ZeroWater). You can boil it too but then you have to wait for it to cool down. Simply leave the water in an open container and the chlorine will evaporate out...not sure how long that takes.
You'll need to repeat this a few times until you get a stable culture. Throw away one cup of starter mixture then add back one cup water and one cup flour that's been mixed together.
When you mix a fresh batch, leave it out on the counter for a few hours. It will double in size so make sure you have a big enough container. Starter over flows are a pain to clean up. From there, put your covered starter in the fridge for 2 or 3 weeks and then rebuild your starter again. I go 4 weeks maximum. Always put a date on the jar so you know when to do the refresh. Starter is like a pet, you have to take care of it and it's a long term commitment.
At 4 weeks, your starter will have a layer of black water on top. In general, don't worry about. The water will contain some alcohol as the yeast give it off as a by-product. When it's time to refresh, just pour off the layer of water. Give it a smell. It will have a sourdough smell. You'll quickly learn what that is. As far as I know I have never had a starter culture go bad. At one time I kept two cultures for white bread and one for rye. Now I just keep one culture for white bread.
When it's time to make your bread, grab 1 cup of starter for your bread and use the other cup to rebuild...one cup starter, one cup water, one cup fresh flour. Keep it simple.
You don't need added sugar for sourdough but some recipes may call for it. Authentic sourdough is much like authentic French baguettes...flour salt and water is all you really need. Baguettes use added yeast instead of the yeast in the starter. FYI, by French law, authentic French baguettes can only contain flour, salt, yeast and water, nothing else.
Over time, your starter, and your sourdough will develop their own special character/taste based on the yeast in your area.
Good luck!
Best Answer
If you've truly gone anaerobic and the smell is off, you are growing things other than the intended cultures...
As a rule, I simply feed mine flour and water. No sugar. The cultures can get along fine with the flour. (I did read in a reputable baking book about adding leftover water from boiling potatoes, for the starches, but I haven't had a chance to try.)
If the smell is off, I would dump and start over in a more breathable container. Your entire goal is to grow the sourdough cultures and let them flourish in their happy environment. And as you've smelled, there is a distinctive scent of happy sourdough. Now if the smell has changed, you lost your scented sourdough and are now growing something else.
I've kept sourdough for over a decade, and it's from a culture that is 84 years old. But if it smells funny (or gets brightly colored mold), he's going down the sink faster than dishwater. It's just the nature of the beast.