The status of a child with regards to desecrating Shabbos on their behalf and giving them medicine:
Having a gentile cook food for him: It is permitted to ask a gentile [See Q&A regarding a Jew] to cook a food for a child [See Q&A regarding the definition of a child] which does not have anything else to eat, as the typical needs of a child has the Halachic status of the needs of a sick person that is not in danger [which was explained above regarding what may be done for him]. [See Q&A regarding if non-food related transgressions may also be done for the child]
Moving Muktzah for a child: It is permitted to feed a child Muktzah [items such as food] even with ones hands if it is impossible to do so in a different way [meaning that no gentile is available].
Summary-The status of a child with regards to desecrating Shabbos on their behalf:
The typical need of a child [which still needs its mother in order to be fed baby foods] has the Halachic status of the needs of a sick person that is not in danger. Thus it is permitted to ask a gentile to cook a food for a child which does not have anything else to eat.
As well it is permitted to feed a child Muktzah food even with ones hands if it is impossible to do so in a different way.
Until what age are the typical needs of a child considered to be like the needs for one who is bedridden?
This matter is disputed in the Poskim:
Some Poskim say that it is dependent on if he still needs to be fed foods by his mother. Thus once the child begins eating like an adult [on his own] then he is no longer considered like one who is ill.
Other Poskim brings that until the age of 9 a child is considered like one who is ill.
Other Poskim rule that the child is only considered like an ill person until the age of 2-3.
Other Poskim rule that the child is only considered like an ill person until the age of 6.
What needs of a child may be done as if he were bedridden?
Some Poskim rule that only with regards to making food for the child is the child considered like a bedridden patient to which a gentile may do all of his needs. However with regards to other matters one may not be lenient any more than any other person.
May a child which is sick be given medicine even if he is not bedridden etc?
Seemingly one may give medicine to any child under Bar or Bas Mitzvah, even if the child is not bedridden, and is above the age mentioned above which considers a child like one who is bedridden.
If a child needs food made for him to eat which involves a Shabbos transgression may it be done through a Jew?
If there is no gentile available then it may be done by a Jew with an irregularity being that the child has a status of a bedridden person. Furthermore if there is worry that delay of the feeding can lead to illness then all transgressions may be done as for a lethally ill patient.
This may only be done to a child not older than the age mentioned above, and only if it is a type of cream which needs to be rubbed completely into the skin and it is not common to leave any of visible cream on the skin. If the cream is common to be left visibly on the skin then it is forbidden to be placed. [However according to Ketzos Hashulchan which argues that there is no smoothening prohibition by soap, seemingly here too it would be allowed even if the cream is visible.]
May one smear oil on a baby for healing purposes [such as he has a rash and the like]?
Yes as a child is always viewed as a non-lethally ill patient of which it is permitted for one to give them medication on Shabbos.
 SSH”K 15/18 writes that all Muktzah items may be moved for a child’s need if no other is available.
 Such as fruits that fell off the tree on Shabbos and the like
 Ketzos Hashulchan 134 footnote 18
 Ketzos Hashulchan 134 footnote 18
 See Shabbos Kehalacha Vol. 1 p. 214
 Ketzos Hashulchan 134 footnote 18; Aruch Hashulchan 328/20
 Minchas Yitzchak 1/78; SSH”K
 Sheivit Halevy, Chazon Ish, Rav SZ”A
 Tzitz Eliezer 8/15-12
 Tehila Ledavid 308/47 and 328/24
 Minchas Yitzchak 1/78
 This opinion seems to be supported by the Halacha above which states that “the general needs of a child are viewed as the needs of a sick person”.
 If the child is young enough to be considered even when healthy like a bedridden person, then definitely it is allowed. However perhaps one can argue that even if the child is above this age he may be given medicine under any scenario so long as he is under the age of Mitzvos based on a ruling in chapter 343 that one may be lenient to feed a Rabbinical prohibited food to a slightly sick child if there is no gentile available to do so and thus so too here since medicine is only a Rabbinical prohibition one may be lenient.
Furthermore one can say that even according to the stringent opinion there [in chapter 343] here they would agree to this leniency because they only hold of a prohibition in cases that the food is in it of itself Rabbinically not kosher. However if it is Kosher in it of itself and just cannot be eaten due to a separate Rabbinical prohibition, such as the prohibition to eat prior to Kiddush, then they too agree that the child may be given food. Thus here too being that the medicine is kosher in it of itself and it’s just a side prohibition that prohibits it, therefore according to all it is allowed to be given to a slightly ill child.
 So is evident from Admur. However those which rule like M”B would forbid a Jew from doing Biblical transgressions.
 Chazon Ish 59/4 brought in Piskeiy Teshuvos 328/16
 Doing so may involve a smearing prohibition, as well as a taking medicine prohibition.
Due to the medicine prohibition it may never be done to one who is not defined as bed ridden or weak in his entire body.
 Minchas Yitzchak 7/20; Piskeiy Teshuvos 328/18 Vetzaruch Iyun from Piskeiy Teshuvos 327/1 which contradicts this ruling here and rules that it is always forbidden.
 As in such a case there is no smoothening prohibition, as is proven from the fact that no smoothening prohibition was mentioned regarding soap. [Daas Torah brought in Minchas Yitzchak ibid].
 Beir Moshe 1/36, brought in Piskeiy Teshuvos 327/1