Does radio communication have to account for the doppler effect?
Would be pretty terrible if that wasn't the case: RADAR wouldn't work!
Do moving objects such as planes and rockets have to account for this,
Yes. Phones in cars and trains, too. Your 5G NR phone of the future operating above 60 GHz, will have to do that at walking speeds, too.
or is ...
I don't believe there is much more to it than that, especially noting that you are aware not to track so fast that you track out amplitude information for modulated signals where the amplitude is used as part of the modulation (so more so for M-QAM than M-PSK). You can also design AGC's with different rise/fall times so can have "fast-attack" and "slow-decay"...