Suggested answer:
(1) Police, shorts, and jeans occur only with plural concord in the corpus. Examples are:
The police are on the way.
The jeans were very tight.
He had no legs and his green shorts were wet and crimson...
Everyone and news occur only with singular concord:
Everyone is sick that this happened.
The news was sensational.
Finally, staff was found to occur with both singular and plural concord:
The staff has assembled near the loading dock.
His own staff were dismayed.
(2) Police, shorts, and jeans can be classified as plural-only nouns (invariable plurals), while everyone is a pronoun taking singular concord only; everyone refers to everybody and everyone in a group of people. Even so, they are still perceived individually or as a totality, so the singular verb form is maintained. News is an uncountable noun taking singular concord only. Staff was seen to occur with both singular and plural verbs; it can be classified as a collective noun. When it occurs with a plural verb form, the emphasis is on the group as consisting of several members, while when it occurs with a singular verb form, the emphasis is on the group as a unit.
(3) No, based on the corpus results, only all accepts both singular and plural concord. (The one occurrence of the combination everyone were shows the subjunctive verb form rather than the plural past tense.)
(4) See the answer under (3) for a discussion of everybody and singular concord. As regards all, it can be used with both plural and singular verb forms. Singular verb forms are used when we are referring to all as a totality of items. Here, all is close in meaning to everything. When it is used with a plural verb form, it refers to the individual members of the group included in all. With all we have a case of notional concord, i.e. concord is determined by the sg/pl idea of the noun.
Read about concord here and here, about plural-only nouns here
Terminology: concord, singular, plural, collective noun, indefinite pronoun, notional concord