Both of them sound okay to me. :)
For your first example:
- 1.) "I have never had a car, nor a bicycle either."
Grammatically, that seems fine. In the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language (CGEL), page 1309, there's a section on "Nor as coordinator", and below is a relevant excerpt from it (note the three examples in [50.ii]) :
Nor as coordinator
Nor appears as a coordinator paired correlatively with neither ([50.i]), or non-correlatively as a variant of or in negative contexts ([50.ii]):
- i.a. [Neither Jill nor her husband] could help us.
- i.b. A good conversationalist talks [neither too much nor too little].
- ii.a. The change won't be [as abrupt as in 1958 nor as severe as in 1959].
- ii.b. No state shall have a share [less than 50% nor more than 70%].
- ii.c. Serious art is not [for the lazy, nor for the untrained].
In [ii] nor could be replaced by (or, which is much more common:
the version with nor perhaps gives added emphasis to the negation. .
. . The difference is that in [i] all the coordinates are marked as
negative, whereas in the non-correlative [ii] the first coordinate
(as abrupt as in 1958, etc.) is not marked as negative within the
coordination itself, but falls within the scope of a preceding
For your second example:
- 2.) "I have never had a car, or a bicycle either."
Grammatically, that seems fine. Your "either" seems to be functioning as a connective adjunct. Here are some examples that might be somewhat similar in some ways:
She wasn't impressed, (and) I wasn't either. -- (CGEL, page 1308)
Kim didn't go and Pat didn't, either. -- (CGEL, page 1305, fn 25)
Kim didn't like it either. -- (CGEL, page 570)
So, it seems to me that both examples are fine.