Can someone help me with finding out when to use -er -es -em, etc. From what I understand, its the ending from diese. For example, in this case Haushalt is Masculine and in the Nominitiv Case. Then, it would be dieser, so -er, which makes it kleiner. Am I correct? If yes, is there a way to remember this?
You are correct. The way I remember it is a two-question process:
You need to know the gender and case of the noun. This corresponds to one of the der/die/das/den/etc. variants. The last letters of the appropriate variant are kind of a 'signal' that needs to be shown somehow in the phrase. Does another word in front of the noun (like an article or similar) show a signal already (i.e. have these or other letters at the end due to declension)? If not, then the adjective needs to have this ending.
If another word in front of the noun does have some kind of signal at the end of it, the adjective doesn't need to show it. It does however need to act in a supporting role, telling us whether the signal shown by the other word is 'normal' or not. So if the signal is any of only der/die/das for a singular noun, we say this is one of the 'normal' signals, and the adjective gets -e at the end. If the signal being shown is anything other than those three (e.g. den/dem/der for dative/die for plurals/a genitive -s) then the adjective gets -en.
Or you can just memorise tables.
No, "er" would be wrong.
"The personal pronoun es replaces a noun used as subject in a sentence with a predicative (with sein). When it is used with the verb sein and a predicative nominative, es can refer to singular and plural nouns of all three genders. The verb agrees with the predicative, not with es"
It's because of declention. This is a nominative noun with an indefinite article (ein), so the adjective ends in -er. It's kinda complicated. Here's some charts. https://en.wikipedia.org/wiki/German_declension#Weak_inflection.5B6.5D.5B7.5D
Comparative is always first. A comparative without an ending (Sein Haus ist kleiner als mein) is still, effectively, an adjective by itself and therefore it takes an adjectival ending much like it would without the comparative. ...I got confused typing this, so I really hope this makes sense to you.