Fiat forum - club

Do you have obligatory test in Czech Republic? They test brakes, suspension, look for corrosion and check emissions (gas analyzer)?
Is it called "STK"? Also, any good garage, professional mechanic, should have a gas analyzer.
Go and check the emissions, don't guess (that you have a rich mixture ...

Filter (1:23): https://www.youtube.com/watch?v=8LhNV0CetJY

Those engines (Grande) are NOT like old FIRE. Now they are interference. You do NOT use "factory marks", no 0 deg. TDC and so on.
You make your own markings or better use locking tools.
Polish forum. Paint marker method. http://fiatpunto.com.pl/topic54365.html
Tools, drawings. http://fiatpunto.com ...

What's the brand of the new alternator?
Denso, FAST, Bosch, other?
New or "new" (reconditioned)?

Bad alternator = empty, weak battery. You should charge it outside the car or do some long trip (like 50-100 km and back home).

Some alternators (regulators) are picky about starting procedure ...

Old (2004) service manual, introduction to Grande model. Italian language. Big PDF file 125 MB.
http://www.fiat-bg.com/Downloads/Books/Grande_Punto/Fiat_Grande_Punto_Service_Manuale.pdf

C514 gearbox family manual, English.
https://drive.google.com/file/d/15OJS6_dKHcPpCRDWebTPtn7LJynIRw8m/view ...