In order to safeguard you from instructing gentiles to do melacha for you on Shabbos, Chazal forbade you to benefit from any melacha or shvus done solely for your sake by a gentile. (ShA 1 – MB 1,2)
If a melacha d’oraisa was done for you, like lighting a lamp, it is ossur for any Jew to benefit. Whereas, if it was merely a shvus, it is forbidden only for the Jew it was done for. (MB 3)
In a case that the lamp was lit for the gentile himself, any Jew may benefit from the light. Let’s say the gentile accidentally put out your light, if he chooses to relight the lamp — it is considered that he lit it for himself. (ShA1 – BeH “l’tzorko”)
Another hetter is when the gentile lights the flame for a Jewish sick person — even if it isn’t life threatening. Anyone can benefit from it. (ShA1)
A child who has a legitimate need is in the above category of a sick person. In Siman 328 we learn that you can instruct a gentile to do melacha for them. Therefore anyone may benefit from the melacha. (Rama 1 – MB 6,7)
If you were to instruct a gentile to put on a lamp in your lodgings, you would be obligated to leave in order not to benefit from the light. If you see the gentile about to put on the lamp, protest and chase him out of the house. If he lit the lamp against your will, warn the gentile not to do that again — but you do not have to leave the room nor turn away from the light. (However, there is room for middas chassidus even in this case.) It is still forbidden to use the lamplight for something you were not able to do when it was dark. If it was a furnace that was ignited, you don’t have to leave the room but do not warm yourself right by it. (ShA 1 – MB 11, 12, 13, 14)
How do you know the intentions of a gentile who put on a light in a room full of Jews and gentiles? If you see him using the light, all Jews may use the light. Also, if the majority of people in the room are gentiles, Jews may use the light. (ShA 2)
The opinion of the Ittur is that you are allowed to instruct a gentile to turn on a light so you can eat the Shabbos meal (Amira l’nochri bimkom mitzva). Most poskim disagree. You can rely on the Ittur only during bain hashmashos Friday or Saturday – or in a case of a mitzva for the public (e.g. to fix a broken eruv). (Rama 2 – MB 24, 25)
You are allowed to send a gentile on an errand in the dark even though he lights a lamp for himself, since you don’t benefit from the lamp at the time it is lit. When he comes back with the lamp, you may also use the lamplight. (MB 27)
Similar to the above: a non-Jewish maid may turn on a light to wash your dishes. Her job is to wash the dishes, but not to put on the light — if she does so, it is for her own convenience. You can even assist her in the washing by the lamp that she lit. (MB 27)
There is a minhag to add on paragraphs of praise to the Shacharis daavening in honor of Shabbos. (Rama)
If you are late to shul do not skip Nishmas and do not skip Hakol Yoducha nor Lokayl Asher Shovas in order to daaven Shemoneh Esray with the tzibbur. If you did skip them, you do not have to go daaven again. (Rama – MB 2, 3 – BeH “uv’chol dovor”)
If, after Shemoneh Esray, you realize that you said the weekday Birkos Krias Shema, you should recite from “… Shevach nos’nim…” and lead into Lokayl Asher Shovas. Say it without a beracha. (Rama – MB 3)