The easy part first. Words like team, police, and United States are collective nouns, which means that they refer to a group of people or things. You will find information about them in grammar books under this heading.

In British English, they can all be followed by a singular or plural verb:

There is a varied programme of events taking place across Scotland, but the Scottish team is always open to new suggestions.

The HSE inspection team were impressed with the level of commitment to health and safety found throughout the Fire service.

Now the police has decided to make us pay another £4,000 plus, for the privilege of entertaining tens of thousands and earning the city a fortune in the process.

The police have 48 hours to object to notices.

The United States is fantastically rich.

The United States have numerous legal cases outstanding.

Actually, the United States is not a particularly good example here, as it is generally followed by a singular verb, even in British English: I had to hunt around in the corpus to find examples with plural verbs. A more typical collective proper noun is BBC:

The BBC broadcasts over 37,000 hours of music a year - an astonishing amount.

The BBC produce education material for school children, informative websites as well as thousands of hours of TV and radio to appeal to the entire country.

In American English it is normal to use a singular verb following a collective noun. In fact, if you look up the entry for government in the British edition of Macmillan English Dictionary, you will find a note telling you that in British English, government can be used with a singular or plural verb, while the entry in the American edition carries no such note. So if you are writing in an environment in which American English is the norm, it is probably best to stick to a singular verb at all times.

As for which form of the verb to use when (assuming you are using British English as a model), this really is up to you. It is probably true that by using a singular verb you stress the collective, unified nature of the group being referred to, while a plural verb can highlight the fact that the group is made up of many individuals. However, it is often hard to identify any such distinction, as the following examples show:

The audience was given the opportunity to comment on the documentary and ask questions of the panel.

The audience were invited to interrupt or stop the sketches at any point to change the outcome of events.

Maybe they should have ended the encore with High, as that would have summed up exactly how the audience were feeling.

Dan particularly appreciated that the audience was discerning and wanted to watch and to listen.

One thing to note is that if you use a singular verb, any pronoun or determiner that refers back to it should be singular, and the same goes for a plural verb:

I have absolutely no doubt the government knows exactly what it is doing.

The team revealed how they felt about his approach and how the 'lockout' affected them and their class.

You can find more on this topic on the Macmillan Dictionary blog.

You are correct to say that whom can be used as a relative pronoun when it is the object of a verb or preposition, as is shown by these corpus examples:

On hearing her name, I asked if she was related to a man whom I had known long ago in Ireland.

‘Really?’ the pretty girl to whom he was talking exclaimed.

Suppose you know a disaster is imminent, but don’t know whom to tell?

However, if you look at the entry for whom in the MED you will find it says: ‘Whom is only used in written English and in formal spoken English.’

In fact, many people nowadays consider whom to be rather too formal to use in everyday writing, as well as in speech, and prefer to use who, as in these examples:

I saw a man who I did not know, checking traps.

Let us know who you spotted and what they were up to, by filling in the form below.

The MED entry goes on:

‘Who is normally used as the object of a verb or preposition, but immediately after a preposition, whom is generally used: the man with whom she lived. It would, however, be more natural to say: the man she lived with.’

That is, we tend to avoid using whom, even when the linguistic circumstances seem to demand it, because it sounds overly formal. This is probably the reason why the author of your coursebook shows a preference for who as both subject and object relative pronoun.

This is a bit of a teaser. I think the pronoun you refer to may be that, since it can be used as a determiner (which you might call an adjective), a pronoun (which acts like a noun, although it isn’t one) and an adverb:

Give me that book (determiner/adjective).
Give me that! (pronoun/noun).
I’m not that bothered (adverb).

This behaves in the same way:
Have you seen this film?
What’s this?
I never thought I could be this happy.