The general construction is referred to as gapping by The Cambridge Grammar of the English Language p1542:
‘Gapping’ is the name given to the construction where the medial
segment of a clause is ellipted when anaphorically retrievable:
[81]
i Kim lives in Perth, Pat _ in Melbourne.
ii Tom will play the guitar and Mary _ sing.
iii A: I will now show you how to make clafouti. B: And I _
custard.
Nor is equivalent to 'and-not' or 'also-not' (p1310), and used in negative contexts.
The below are some examples of the construction being used in formal contexts.
They can not be expected to subsidise us, nor we them (House of Commons British Parliament; Mr Gerald Reynolds; 1965)
the Catholics understand that you can not trust them, nor they you, this fact is at the bottom of all your difficulties (House of Commons
British Parliament; Viscount Howick; 1844)
We do not regard Commonwealth citizens as aliens; nor they us (House of Lords British Parliament; Mr Joseph Garner; 1971)
White House aides, adapting to his mellow managerial style, seldom prod the President, nor he them. Instead, Reagan waits for an
amiable consensus to develop among his advisers, who work within the
boundaries of Reagan's ideology. (Time Magazing; A View Without Hills
or Valleys; Kurt Andersen; 1984)
When a man and wife have their minkuyu in the house they may not put their hands into such a container; the husband may not touch the
wife's nkuyu, nor she his. (ACAD: African Arts; Northern Kongo
ancestor figures; 1995 Spring)
Halpin and Beaumont do not enter Lamont's world, nor he theirs. (ACAD: Style; A postmodern challenge to reference-world construction:
Gilbert Sorrentino's Mulligan Stew; Charles, Mayl 1995 Summer)
We'd used condoms for the entire three or four months we'd been seeing each other, so I was pretty sure he hadn't infected me -- nor
I him. (MAG: Essence; Facing AIDS; Lewis-Thornton, Rae; 1994
December)
The first example under your third question is ok, but the other two are deficient in that they contain nominals (NPs lacking determiners) with common count singular nouns as head which cannot function as objects like Germany and TV do, so we must add a determiner.
*She went to Germany, and I (went to) park.
She went to Germany, and I the park.
*He was watching TV, and I (was watching) hamster cage.
He was watching TV, and I the hamster cage.
