Can't seem to get my rye sourdough starter really going
Your problem likely has little to do with moisture or temperature -- starters can actually be established in a wide variety of conditions, from a dry dough ball to very wet, and can from very cool temperatures to quite warm ones.  It may take more or less time to establish in some conditions, but your parameters sound well within range.  I would check the "oven with light on" temperature if possible, just to see what it is, but I doubt it's high enough to cause problems.  (You probably don't want to go much above 85-90F when trying to establish a wild-yeast starter, and a little lower than that is probably ideal.)

As FuzzyChef said, it's possible that chlorinated water or old flour can cause problems.

But my guess from your description that your starter rose significantly with greater feeding is that it's simply starving.  Once yeast begins to be established (usually 3-4 days after beginning a starter, with 24-hour feedings, but can be anywhere from about 2 to 7 days depending on conditions), you should increase the frequency of feedings.  I generally prefer twice per day, but it depends on temperature and amount fed.  For 1:1:1, every 12 hours or so is reasonable.

The problem with letting the starter sit for 24 hours at a time while trying to get established is that the yeast usually have peak growth pretty early on (within a few hours of a feeding), but after several hours, mostly you're just growing lactic acid bacteria, which makes the starter acidic ("sour").  That's good, up to a point.  The problem is when the starter begins to get too sour, yeast die off, and they run out of food without feedings.

A 12-hour cycle might be enough of a shift to jumpstart the yeast growth and catch them before they start dying off every day.  (Some people with so-called "sluggish" starters or starters that have become too acidic will even try more frequent feedings for a limited time.)  I think at this point you might have a relatively small amount of yeast that just isn't growing much from day-to-day.  Try feeding more frequently for several days and see what happens.  For a 100% hydration starter (which you have), you usually want to see a starter that can consistently double or even triple in size within 2-4 hours after a feeding before you start trying to bake bread with it.