For an explicit present progressive, it is generally recommended to use être en train de…, in which case “I think they are being used”, in the sense of “I think they are in the process of being used”, would be rendered as je crois qu’ils sont en train de s’utiliser, or je crois qu’ils sont en train d’être utilisés for an explicit emphasis on the “beingness” of the utilisation.

1. I think they are being used: je crois qu’ils sont utilisés

It seems, first, that this sentence in English expresses typically the fact that people in question are being abused in some way.If you wanted to say that someone is using something you’d say more naturally “I think someone’s using them.” or “I think they are using them.” or “I think there are people using them.”

I think they are being used. — Je crois que quelqu’un les manipule. (1)

I think someone’s using them. — Je crois que quelqu’un s’en sert or (entirely equivalent) Je crois que quelqu’un est en train de s’en servir. (2)

2. they were being told not to enter the house when he collapsed: “ils étaient donné l’ordre de ne pas entrer la maison quand il a eu une perte de conscience.”

The translation of the first verb form can only be “On leur donnait” or “on était en train de leur donner”; moreover the context as elicited from the English sentence is a very unlikely one: to give an order takes very little time and therefore can’t be expressed by a progressive form; What would you say of “He was eating bread when suddenly one of his teeth was breaking.”? (broke).¹ Next is the translation;

On était en train de leur donner l’ordre de ne pas entrer dans la maison quand il a eu une perte de connaissance. (3)

3.It hasn’t been used in years: ça fait longtemps que ça n’est pas été utilisé. (4)

That one is almost correct except for the auxiliary verb: it’s “avoir” not “être”; “hasn’t been used” is not a progresive tense; it is a passive voice present perfect (They have used it for years.); prog: not used, it would be “it has not been being used”.

(…que ça n’a pas été utilisé)

4. • (1) Je crois qu’il sont manipulés par quelqu’un.

   • (2) (3) (4) No passive form

¹ _{This is a correction made after comments; the assertions, downright wrong or insufficiently precise, have been stricken rather than removed so as not to render the flow of comments meaningless. In the light of the comments (OP and Janus Bahs Jacket) I conceive the progressive form as fully justified in the present context, that as a consequence, essentially, of the characterisation of the span of time over which the action is acknowledged as having to be short being only a very relative notion. The example given (broken tooth) is correct but its context does not match.}