Hello,
I suffer from having my non-fretting fingers 'float' away from the fretboard when they are not pressing down on a fret, with my pinky forming the classic 'cup of tea' pose in particular. I was wondering if I could expect them to become more controllable and hover above the fretboard if they were stronger as a result of doing some serious trill exercises? I'm a beginner and I have terribly weak and undeveloped fingers. Has anyone experienced this? Thanks.

Spiders & ladders will fix that. Particularly spiders.

Another good thing to try is to reduce the pressure you use with your left-hand fingers. If one finger is pressing down too hard, it can cause another to lift too far from the fretboard when you release it. Try fretting a note and then picking it. Then keep reducing the pressure you're using to fret the note while continually picking. Do this until the note goes dead. You need no more pressure than just a bit more than this to fret the note.

Also, try and play really slowly so you can accurately control your left hand fingers. I tried this for a couple of months last year and it really helped with my floating pinky - still not perfect, but much better than it was previously. Once the pinky has lifted too far off the fretboard, it's too late to fix it - so you have to be consciously aware of it's motion when your training it and ensure it lifts no further than necessary.

Pebber is right...

Trills help too...

Concentration...