Manual First Gear Problem
 

Hello guys. I'm new to this whole forum thingy.
There are a couple of things WRONG with my vehicle but today I'm going to ask about the transmission problem I have :(
I am a total Noob when it comes to automobile.. anyhow I drive a 06 Mazda3 GT Manual transmission.
My ex-girlfriend (newby manual driver) accidently shifted into the First gear while going 100km/h. I don't know how she managed to shift into the First gear but when she did, there was grinding noise and the clutch was no longer available (VERY LOOSE). I somehow managed to put into gears and drove it to the tranny shop and got the clutch replaced. HOWEVER, even after replacing the clutch, my car SOMETIMES makes grinding noise when I shift into First gear.
I have been getting some opinions that the problem with my vehicle is either the Synchronizer or chipped in First gear.
My understanding is that to fix either the synch or chipped gear, I'd have to take the tranny apart.
So now my REAL question is... what would be the cheapest way to fix this problem? and how much would it cost?

How many miles and on what year?

I would guess the synchro / chipped gear sounds about right - though that shouldn't happen if the trans was shifted from a high speed into a low gear. If the clutch is disengaged there may be some grinding, but otherwise the trans should not have an issue.

Cheapest fix... ignore it - whatever is wrong is a trans issue, and once it breaks the rest of the way just replace the trans. And practice starting in 2nd gear to prepare for when 1st finally dies (if it dies)