Learn Danish: Vi tror at dette tårn er nyest. - We believe that this tower is newest.

Because it is being used predicatively (after the verb er) and without a definite article. When stating that something 'is newest' in a general or indefinite way, regular adjectives just take the -est suffix. If we wanted to say 'the newest tower', it would become definite and need the -e ending: det nyeste tårn.

Could I use synes or tænker instead of tror here?

Not naturally. You use tror when you are guessing or expressing a belief about a factual thing (like which tower was built most recently). You use synes for personal subjective opinions (like whether the tower is beautiful), and tænker for the active process of using your brain to think.

Why is the word for 'this' dette and not denne?

The noun tårn (tower) is a neuter word, meaning it takes et instead of en. The demonstrative word for 'this' has to match the gender of the noun, so we must use the neuter form dette. If it were an en word, like en bygning (a building), you would use denne bygning.

Can I drop the word at just like we sometimes drop the word 'that' in English?

It is possible to drop at in very casual spoken Danish, but it is much safer and more natural for learners to keep it. In written Danish, at is almost always kept to clearly link the main sentence to the secondary clause.