filter sqlite query based on counts of pairwise interactions

I am trying to filter a somewhat involved sqlite3 query using a pairwise association table. Say I have these tables (where pet_id_x references an id in table pets):

[pets]
id | name    | animal_types_id | <additional_info>
1    Spike     2
2    Fluffy    1
3    Whiskers  1
4    Spot      2
5    Garth     2
6    Hamilton  3
7    Dingus    1
8    Scales    3
.    .         .
.    .         .

[animal_types]
id | type
1    cat
2    dog
3    lizard

[successful_pairings]
pet_id_1 | pet_id_2
1          4
2          4
2          8
3          2
3          4
4          5
4          6
4          7     
5          6
5          7
6          7
.          .
.          .

A toy example for my query would be to get the names of all dogs which meet certain constraints (from columns within the pets table) and have > 2 successful pairings with other dogs, resulting in:

name  | successful pairings
Spot    6
Garth   3

As per the above, the ids need to be combined from pet_id_1 and pet_id_2 in successful_pairings to get the total count of pairings involving each pet id.

I am new to sql syntax, and am having trouble chaining queries together to filter based on conditions distributed across multiple tables.