Group 1)

• Could you dress the children?
  Correct.
• Could you put on the children?
  Sounds like you are going to wear the children, not dress them. You could stretch that to Could you put the clothes on the children? which would clarify the meaning, but would still leave it awkward. I'd stick to the first version.

Group 2)

• I bathed her and dressed her in her new clothes.
  Correct, if a little clumsy due to the many repeats of the word her. Might be ameliorated by using her name in the first instance, I bathed Susan and dressed her in her new clothes
• I bathed her and put her new clothes on her.
  You're right that this one is very awkward. It's correct but just doesn't flow nicely. If you replaced the first her again with a name, that would help a bit, but the only way to remove the double her at the end would leave it sounding like you were going to try on her clothes yourself I bathed Susan and put her new clothes on. I'm not sure they'd suit you, or even fit ;-)

Group 3)

• I can’t bend. Could you dress me in my socks?
• I can’t bend. Could you put my socks on me?
  I'd just go for I can’t bend. Could you help me put my socks on?
  That would leave no confusion as to who was to wear the socks, nor who needed assistance; neither would it strictly request they did all the work, but that you would do the best you could whilst they assisted.

Group 1)

• Could you dress the children?
  Correct.
• Could you put on the children?
  Sounds like you are going to wear the children, not dress them. You could stretch that to Could you put the clothes on the children? which would clarify the meaning, but would still leave it awkward. I'd stick to the first version.

Group 2)

• I bathed her and dressed her in her new clothes.
  Correct, if a little clumsy due to the many repeats of the word her. Might be ameliorated by using her name in the first instance, I bathed Susan and dressed her in her new clothes
• I bathed her and put her new clothes on her.
  You're right that this one is very awkward. It's correct but just doesn't flow nicely. If you replaced the first her again with a name, that would help a bit, but the only way to remove the double her at the end would leave it sounding like you were going to try on her clothes yourself I bathed Susan and put her new clothes on. I'm not sure they'd suit you, or even fit ;-)

Group 3)

• I can’t bend. Could you dress me in my socks?
• I can’t bend. Could you put my socks on me?
  I'd just go for I can’t bend. Could you help me put my socks on?
  That would leave no confusion as to who was to wear the socks, nor who needed assistance; neither would it strictly request they did all the work, but that you would do the best you could whilst they assisted.

Group 1)

• Could you dress the children?
  Correct.
• Could you put on the children?
  Sounds like you are going to wear the children, not dress them. You could stretch that to Could you put the clothes on the children? which would clarify the meaning, but would still leave it awkward. I'd stick to the first version.

Group 2)

• I bathed her and dressed her in her new clothes.
  Correct, if a little clumsy due to the many repeats of the word her. Might be ameliorated by using her name in the first instance, I bathed Susan and dressed her in her new clothes
• I bathed her and put her new clothes on her.
  You're right that this one is very awkward. It's correct but just doesn't flow nicely. If you replaced the first her again with a name, that would help a bit, but the only way to remove the double her at the end would leave it sounding like you were going to try on her clothes yourself I bathed Susan and put her new clothes on. I'm not sure they'd suit you, or even fit ;-)

Group 3)

• I can’t bend. Could you dress me in my socks?
• I can’t bend. Could you put my socks on me?
  I'd just go for I can’t bend. Could you help me put my socks on?
  That would leave no confusion as to who was to wear the socks, nor who needed assistance; neither would it strictly request they did all the work, but that you would do the best you could whilst they assisted.

Group 1)

• Could you dress the children?
  Correct.
• Could you put on the children?
  Sounds like you are going to wear the children, not dress them. You could stretch that to Could you put the clothes on the children? which would clarify the meaning, but would still leave it awkward. I'd stick to the first version.

Group 2)

• I bathed her and dressed her in her new clothes.
  Correct, if a little clumsy due to the many repeats of the word her. Might be ameliorated by using her name in the first instance, I bathed Susan and dressed her in her new clothes
• I bathed her and put her new clothes on her.
  You're right that this one is very awkward. It's correct but just doesn't flow nicely. If you replaced the first her again with a name, that would help a bit, but the only way to remove the double her at the end would leave it sounding like you were going to try on her clothes yourself I bathed Susan and put her new clothes on. I'm not sure they'd suit you, or even fit ;-)

Group 3)

• I can’t bend. Could you dress me in my socks?
• I can’t bend. Could you put my socks on me?
  I'd just go for I can’t bend. Could you help me put my socks on?
  That would leave no confusion as to who was to wear the socks, nor who needed assistance; neither would it strictly request they did all the work, but that you would do the best you could whilst they assisted.