Can I use a 9 speed shifter with a 7 speed drivetrain, with good
shifting and without needing complex adaptations?
Kinda, maybe, sort of... its complicated
You could use a Shimano 7 speed derailleur with a 9 speed shifter, if you also used a wheel with a 9 speed cog. The 7 and 9 speed Shimano derailleurs use the same derailleur ratio of 1.7 (amount of derailleur movement for a given cable pull) despite the two systems having different cable pulls (2.9 and 2.5 respectively) and sprocket pitches (5.00 and 4.35 respectively).
If you used a 7 speed cassette, with a 9 speed shifter and 7 speed derailleur you would not pull enough cable and the shifts will get mangled.
If you are currently on 7 Speeds, this means you have a 7 speed wheel that will most likely NOT take a 9 speed cassette. This would mean you would need to buy a new rear wheel as well as a new 9 speed cassette in order to keep you 7 speed rear derailleur. You would also need a 9 speed chain (which is narrower) to mesh with the 9 speed cassette, which may or may not work okay with your 7 speed chain rings.
For pull ratios see http://blog.artscyclery.com/science-behind-the-magic/science-behind-the-magic-drivetrain-compatibility/