tiisai setai ni site ha, sinseisyo ga ooi.

Not exactly. 世帯 means household: the people living together as one unit.

That can be a family, but it is a more official/statistical word than 家族.
So in this sentence, 小さい世帯 means a small household, usually meaning a household with few members.

Here, 小さい does not usually mean physically small in size.

With 世帯, it often means the household is small in scale, especially that it has few people. So:

• 小さい世帯 = a small household
• often understood as = a household with a small number of members

Japanese often allows this kind of shorthand when the meaning is clear from context.

にしては means something like:

• for
• considering
• in spite of being
• given that it is

It also carries the nuance of unexpectedness.
So 小さい世帯にしては、申請書が多い means:

• For a small household, there are a lot of application forms.
• Considering it’s a small household, the number of application forms is high.

The speaker is saying: I would normally expect fewer, but there are more than expected.

Because にしては is a set grammar pattern.

With a noun, the pattern is:

• noun + にしては

So:

• 小さい世帯にしては

is the correct structure.

としては is a different expression. It means more like as / for / in the role of and does not have the same “contrary to expectation” feeling.

Compare:

• 学生にしては日本語が上手だ
  = For a student, their Japanese is good.
  (better than expected)

This is the same pattern as your sentence.

Because 多い works like an adjective meaning many / numerous, and the thing that is many is commonly marked with が.

So:

• 申請書が多い = there are many application forms

This is different from English, where we might think of many as just a determiner before a noun. In Japanese, 多い is often the predicate itself.

Other examples:

• 人が多い = There are many people.
• 仕事が多い = There is a lot of work.

So 申請書が多い is completely normal grammar.

Because 多い usually does not directly modify a noun in the same way English many does.

So 多い申請書 is usually unnatural if you mean many application forms.

Instead, Japanese normally uses:

• 申請書が多い = there are many application forms
• 多くの申請書 = many application forms

So in your sentence, 申請書が多い is the natural predicate form.

It sounds like a judgment based on expectation.

Because of にしては, the speaker is not only saying that there are many application forms. They are also saying:

• That is more than I would expect for a small household.

So the sentence has a mild nuance of:

• surprise
• contrast
• evaluation

Without that nuance, a more neutral sentence might be:

• 申請書が多い。 = There are many application forms.

But 小さい世帯にしては adds: for such a small household, that’s a lot.

Usually 小さい世帯 is more straightforward and natural here.

Both 小さい and 小さな can sometimes appear before nouns, but they are not always interchangeable in every case. In ordinary factual descriptions like this one, 小さい世帯 is the safer and more standard choice.

So for learners, it is best to understand this sentence as a normal い-adjective + noun pattern:

• 小さい世帯 = small household

In にしては, you should learn the whole thing as one grammar expression.

Historically, the は is the topic particle, but in practice, for learners, it is best to treat:

• にしては

as a single set phrase meaning for / considering / compared with what one would expect from.

So instead of analyzing it word by word every time, it is more useful to memorize:

• N にしては
• for N / considering N, unexpectedly...

That will help you understand and use it more naturally.