I tried recreating your situation and I wasn't able to get Oracle to use the indexes wisely. I'm sure there's some smart way to do it. But if no one else here can figure it out, below is the dumb, ugly way.
Since you're only getting a certain number of levels you can manually create a connect by. Get the first level, union that to the second level (which gets results from a copy of the first query), union that to the third level (which gets results from a copy of the second query), etc. I only did three levels here, but you can copy and paste to make the fourth. It's harder to use since the original id is repeated so many times, but it's super fast (0.005 seconds on my machine with 1.6 million records.)
--Original animal
select '101' original_animal, animal_id, line_id, sire_animal_id, dam_animal_id, 0 "level" from animals where animal_id = '101'
union all
--Parents
select '101' original_animal, animal_id, line_id, sire_animal_id, dam_animal_id, 1 "level" from animals
where animal_id = (select sire_animal_id from animals where animal_id = '101')
union all
select '101' original_animal, animal_id, line_id, sire_animal_id, dam_animal_id, 1 "level" from animals
where animal_id = (select dam_animal_id from animals where animal_id = '101')
union all
--Grand parents
select '101' original_animal, animal_id, line_id, sire_animal_id, dam_animal_id, 2 "level" from animals
where animal_id =
(
select sire_animal_id from animals
where animal_id = (select sire_animal_id from animals where animal_id = '101')
)
union all
select '101' original_animal, animal_id, line_id, sire_animal_id, dam_animal_id, 2 "level" from animals
where animal_id =
(
select dam_animal_id from animals
where animal_id = (select sire_animal_id from animals where animal_id = '101')
)
union all
select '101' original_animal, animal_id, line_id, sire_animal_id, dam_animal_id, 2 "level" from animals
where animal_id =
(
select sire_animal_id from animals
where animal_id = (select dam_animal_id from animals where animal_id = '101')
)
union all
select '101' original_animal, animal_id, line_id, sire_animal_id, dam_animal_id, 2 "level" from animals
where animal_id =
(
select dam_animal_id from animals
where animal_id = (select dam_animal_id from animals where animal_id = '101')
);