Some light fixtures can adjust light level using a gauge, but I'm renting and the utilities are beyond my control.
There are many options for smart lights these days (Philips Hue for example) which offer levels of dimming via an app on your phone.
my idea was to simulate a shoot-through umbrella by taping white printer paper over the lights (they are flat ceiling lights that don't protrude)
You'd be better off with a thin sheet. No matter what you do, make sure you're not risking a fire. Switch to efficient LED's and make sure there is adequate airflow. Instead of completely covering the light at the ceiling level, you should place the scrim at least a few inches away to allow for airflow.
The light seems very harsh and it shows in the photos.
A straight bulb is a point light source or hard light. This is because none of the light is being scattered but instead is all coming from essentially the same angle. You need modifiers to scatter the light.
If I turn on the adjacent room's lights, the result is too dark
Then your exposure setting is wrong. What you probably mean is that there isn't enough light to allow for hand-holdable shutter speeds. Well, get a tripod. It takes a good bit of light for a photo and normal indoor bulbs are hardly adequate. Indoor lighting is good enough for us humans to see but it is considered difficult lighting for photography simply because of the lack of light.
While this is definitely not cost-prohibitive, it's somewhat effort-prohibitive (ceiling is a bit high and I'm a bit short).
Your post is coming off as very lazy. Get a ladder.
Anything you are going to do in DIY lighting will require more effort than taping a sheet to the ceiling. Are you sure you really want to do this?
Are there any other DIY methods for more control over home lighting I can exhaust before I go for actual equipment?
At the end of the day, you're the photographer and the lighting is up to you. You can get by with table lamps with homemade snoots and scrims/lightboxes for smaller objects - but the tradeoff is going to be the need to use a tripod and a difficult time controlling your ratios (unless you've got dimmable bulbs).
At some point, if you want to take your studio lighting to the next level, then you need to get yourself a strobe or speedlight.