English grammar question

Both 1 and 2 is correct.

1. He didn’t get up or open the door. - Grammar is no mistake, 不過用and 好過or

2. He did(n’t) get up (nor) open the door. - Grammar is no mistake too.
nor 除咗(用在neither之後), 也可(用在not,no,never之後)
因此無錯!

請上yahoo字典看,證明我的說話
http://hk.dictionary.yahoo.com/search.html?q=1&s=nor

nor
conj.
1. (用在neither之後)也不
I have neither time nor money for pop festivals.
我既沒時間也沒錢來參加流行音樂節。
2. (用在not,no,never之後)也不
The story is not interesting nor instructive.
這個故事沒有味道,也沒有教育意義。
I have never spoken nor written to her.
我跟她從來沒說過話,也沒寫過信。
3. (用在句首,句子須倒裝)也不
You do not like him, nor do I.
你不喜歡他,我也不喜歡。

這兩句都是正確，在not之後可以用or或nor，但用nor的語氣會較強
而且nor多用於有停頓之後
如：She did not phone that day, nor the next day比...that day or the next day更強調）
如：He cannot read nor write的語氣比He cannot read or write強得多
雖然兩句文化都沒有錯，但第一句比較好，因為他既然沒有起身，自然就不能開門，所以無需強調！

希望幫到你！

Both sentences are wrong. It should be:

Neither did he get up nor open the door.