May God bless you for rescuing!
No, f eline pine does NOT clump in the GI tract like litter. He would have had to ingest quite a bit to form a blockage but it could make him nauseated, YES.
Add a half a pepsid a/c tablet to something he will eat right up.
Try a small amount of boiled chicken with all fat drained off, chop meat , or baby food with no onion, beef, turkey, etc
Go very slow so he does not throw it up.
Dogs get dehydrated and hypoglycemic very easily from vomiting.
Add unflavored pedialyte to his water 50/50 to hydrate him and balance his electrolytes.
Put a drop of pancake syrup on his tongue to raise his blood sugar in case he is a bit hypoglycemic. That should perk him up.
Do not allow him to drink a lot at a time If he throws up water, he can lick ice
Keep him on that very light diet of boiled chicken or baby food for 48 hours after the stool is normal.
Also give him one teaspoon a day of plain yogurt to keep "good" bacteria in his system.
If he develops any abdominal pain, bloating, constipation or cannot keep anything down he needs to be seen by a vet.
Please let me know how he is doing. I will always be here for you
I hope your baby feels better very soon..