Learn French: L'invité est censé proposer un toast. - The guest is supposed to propose a toast.

Yes, censé acts like an adjective here, so it must agree in gender and number with the subject. Since l'invité is masculine singular, it is censé. If the guest were female (l'invitée), it would be censée, and plural subjects would take censés or censées.

Do I need a preposition between censé and the infinitive proposer?

No, the construction is strictly être censé followed directly by an infinitive, with no preposition like de or à in between. The English word 'to' is already covered by the French infinitive form proposer.

I've seen the word sensé before. Is it interchangeable with the censé used in this sentence?

No, although they sound exactly the same, they have completely different meanings. Censé means 'supposed to' or 'expected to', whereas sensé means 'sensible' or 'having good sense'. Mixing them up is a common spelling mistake even for native speakers!

Could I use a different verb instead of proposer for making a toast?

Yes, while proposer un toast is perfectly correct, it is also very common to say porter un toast (literally 'to carry a toast'). Both are widely used by native French speakers.