#8. R, 64 bytes

.0->z;x=readLines();y=Vectorize(utf8ToInt)(x);any(grepl("->",x))

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Satisfies:

2. The first character is a .
3. It contains an e
4. Its length is even
5. Its length is a perfect square
6. It contains an a
7. It contains a > character
8. Contains the exact sequence -> in one of its lines.

8. R, 64 bytes

.0->z;x=readLines();y=Vectorize(utf8ToInt)(x);any(grepl("->",x))

Try it online!

Satisfies:

2. The first character is a .
3. It contains an e
4. Its length is even
5. Its length is a perfect square
6. It contains an a
7. It contains a > character
8. Contains the exact sequence -> in one of its lines.

#8. R, 64 bytes

.0->z;x=readLines();y=Vectorize(utf8ToInt)(x);any(grepl("->",x))

Try it online!

Satisfies:

2. The first character is a .
3. It contains an e
4. Its length is even
5. Its length is a perfect square
6. It contains an a
7. It contains a > character
8. Contains the exact sequence -> in one of its lines.

#8. R, 64 bytes

.0->z;x=readLines();y=Vectorize(utf8ToInt)(x);any(grepl("->",x))

Try it online!

Satisfies:

2. The first character is a .
3. It contains an e
4. Its length is even
5. Its length is a perfect square
6. It contains an a
7. It contains a > character
8. Contains the exact sequence -> in one of its lines.