>1. 花を持って、たま子先生に会いに行って、元気づけてあげて下さい。
>Does あげて refer to doing something for the writer, or doing something for たま子先生? I think because it's あげて it should be the latter, but I'm not certain.
「あげて」 here means "to do something for", so the latter.
It may be helpful to remember to distinguish 「~に~してあげる」 (do ~ for ~) from 「~してくれる」 (do ~ for me). If the writer had meant the former, he/she would have written 「たま子先生に会いに行って、元気づけてくれますか」 (would you please go meet Mrs. Tamako and raise her spirits on my behalf), or something in that vein, rather than 「元気づけてあげてください」 (please raise her spirits for her).
>2. This is the start of an email from a boy to his grandmother.
>おばあちゃん、元気ですか。僕は元気で学校に行っています。
>Is 僕は元気 being presented as a reason for 学校に行っています, or are the two things separate?
I don't think there is meant to be a direct causal relationship between 元気 and 学校に行ってきます, but the two are not unrelated either.
I think what's throwing you off is the 「で」, which you might have confused as an abbreviation of 「~なので」. It might help to know that 「元気で~しています」 is a common formality in Japanese letters and is also often written interchangeably as 「元気に...しています」 (I am doing ... in good health).
Breaking it down, you could say that 「元気で/に」 is supposed to characterize the manner in which he goes to school, i.e. the writer's condition as he engages in the activity of going to school.
>3. 僕は日本の踊りは小さい時に盆踊りをしただけですし ・・・ 今まで僕はブレイクダンスしかしたことがなかった・・・
>a) I feel I am missing something here because to me this looks contradictory: the writer did 盆踊り >when young, but has only ever done breakdancing??
>b) Also, does だけ mean "only" in the sense that the speaker did not do any other form of dance when small?
I don't know if you've ever participated in 盆踊り, but it is something that people gather to do once or twice a year at best, and requires minimal training to participate in. Since the passage overall is speaking about the writer's dancing experience, the implication is probably that 盆踊り doesn't really "count" as experience in dancing.
As for your latter question, 「~しただけ」 here is used to emphasize the limited nature of one's experience/actions/contributions, etc. For example, it is commonly used in such phrases like 「僕は手伝っただけです」 (I only helped out), which emphasizes that the extent of one's contribution was limited, and that someone else had done most of the work.
>As for Japanese dance, my experience is limited to the
bon-odori I did when I was young ...
>I've only ever done break dancing before ...
>4. チェンさんはお母さんを手伝いっています。
>Is this a typo for 手伝っています?
Probably yes, but it should be pointed out that it could also be a typo for 「手伝いにいっています」; the meaning would be roughly the same in both cases, although the latter would imply a commute whereas the former would not.
I hope that helps.