Most US churches say 'Our Father, who art'; many UK churches still say 'Our Father, which art,' which is the older form (from the oldest versions of the Book of Common Prayer). A better translation than either is the ecumenical version that simply starts 'Our Father in heaven,' now used in some churches (or some services) in both the US and the UK.
This is a perennial topic, so it would help if you would read some of the many old discussions first, e.g.,related discussion: deren - "whose" oder "of which"related discussion: Usage of "whose"related discussion: deren Entwicklung... - the development of wh...related discussion: deren
There are also a lot of old threads on 'deren' and 'dessen,' though not all of them were well answered.
The short answer: In the subject and object cases, of course we distinguish between 'who(m)' for people and 'that' for things (although in casual speech you will also hear 'that' for people).
However, in the possessive case, it's a myth that 'whose' can't be used for things. 'Of which' is preferred only in extremely traditional BE (one example cited in the last link above: legal and patent texts). In fact, 'whose' is often less clumsy, is widely used for things in all registers in both AE and BE, and there are some cases such as 'an idea whose time has come' where it would really be silly to use anything else.
I'm not sure exactly what you're getting at in #2. There is a 'which/that' issue, but it's a completely different topic. There are also lots of old discussions on it in the archive.
A short review: It's mostly an issue of style, not grammar. But in AE especially, many writers still follow the rule given in Strunk & White: 'That
is the defining, or restrictive pronoun, which
the nondefining, or nonrestrictive. See Rule 3 ['Enclose parenthetical expressions in commas'].'
So if you want to follow this rule (which originally came from Fowler, writing on BE), it's true that you should only use 'that' in a restrictive (defining) clause:
The ostrich that I saw was brown.
The ostrich I saw was brown.
If you use BE and don't choose to follow Fowler's style rule, you can use 'which' in a restrictive clause:
The ostrich which I saw was brown.
But whatever you do, you can't use 'that' in a nonrestrictive clause. If the clause is not defining, not necessary to the meaning of the sentence, but only an extra or parenthetical piece of information, set off by commas, you must use 'which.'
A good rule of thumb for German speakers: Never put a comma before 'that.' If it needs a comma (because the information is not essential to the sentence), use 'which.' Last example:
Ostriches, which are native to Africa, are the only living species of their family.
*Ostriches, that are native to Africa, are ...