1998 Corolla with remote start. It wont work and

You probably have it out of time , marks 180 degrees out . Check your timing marks .

Do you know if the timing belt is ok ? Will the engine crankshaft turn ?
Have you checked for power at the starter ?

check the fuses again, sometimes they blow right after you install them.
there may be a short in the wiring causing them to pop.
check fuses again and let me know

IF THE BATTERY IN THE REMOTE IS GOOD THEN THE CAR HAS TO TAKEN TO THE DEALER TO BE REPROGRAMMED. ALOT OF TIMES I RECOMMEND JUST GETTING AN AFTERMARKET ALARM SYSTEM. BY THE TIME THE DEALER FINISHES GIVING U THE SHAFT ITS ALOT CHEAPER AND U GET MORE FOR YOUR BUCK

It might be your anti-theft alarm system. When the car doesn't start, is there a ticking sound inside the car? If so, hit the unlock button a few times on your keyless entry remote, until the ticking stops, then try to start the car. Hopefully this helps.

if the battery is good and there that sound you have problems with the starter motor

from outside the car make sure all doors a closed completely, make sure hood and trunk are also closed. lock the vehicle with the remote, unlock it and repeat, lock and unlock. if it still wont start, put the key in the driver door lock, lock and unlock the door three times with the key, then try to start.

I had the same issue.
contacts inside the starter are worn out. Try to find replacement contacts $15 and install yourself or replace starter$300

the actuator might have been disconnected from the lock, its attached inside the door, with a metal rod that clips into the actuator, check it out, might be a easy fix, good luck and god bless, please rate me!

I had the same problem, i had to remove the whole alarm system to get my car to start, when you unattatch the battery cables, the alarm notices the battery drain and sets the alarm off, i have no idea how much the dealership would charge for a new remote, but im sure its pricey, i would recommend just removing the alarm system, temporarily, until you can get a new remote. Good luck and god bless