Learn Russian: Сколько весит этот словарь?

The verb весить (used here) means 'to have a certain weight'. It only describes how heavy the subject is. If you want to talk about the physical action of weighing an object (like a cashier weighing apples at a store), you must use the transitive verbs взвешивать (imperfective) or взвесить (perfective).

Словарь ends in a soft sign (-ь). How can I tell that it is a masculine noun?

Nouns ending in the suffix -арь (like словарь, календарь, or пекарь) are masculine. In this sentence, the masculine demonstrative pronoun этот is a perfect clue! If the noun were feminine, the sentence would use эта.

How would I answer this question with 'It weighs one and a half kilograms' using this lesson's grammar?

You would answer: Он весит полтора килограмма. You use полтора (rather than полторы) because килограмм is a masculine noun. When полтора is the object of the sentence, the noun following it must take the genitive singular form, which is килограмма.

Could I change the word order and ask Сколько этот словарь весит?