Daiwoo Lanos 1999. Have to rev high when taking off in 1st gear or engine will die and miss. when driving it runs well so long as the throttle is only pushed to above idle and increased slowly.

Is the shifter real loose and floppy?
If it is you may be starting out in 3rd gear instead of 1st!
Remove the shifter boot below the shifter look at the rods as you shift 1 thru 5 and Rev! Note the rod position as you shift thru the gears. You can do this while sitting idle, not driving!
If while tring to get it into 1st, take your hand and push the left rod farther forward has you move it into 1st gear! If you are able to get it into 1st, try driving forward, if it doesn't stall, chances are you need to replace the shifter do to the worn out Nylon Bushings in the shifter! If this doesn't help, take a look under the hood and see if the shifter cables are attached to the top of the shifter tower! Some times the clips holding the shifter cable to the trans come off and one of the cables will slide making it impossible to get it into 1st gear!
Hope this helps!
if you need photos of what I'm trying to explain, please reply back as soon as possible.

There is no such thing as a new engine for a used car. Are you referring to a rebuilt or re-manufactured engine? If so you can find one online or talk to you're local repair shops.

I would check the engine coolant temperature switch 1st.

when checking the transmission fluid always check it while vehicle is running, if you overfill the transmission YOU WILL BURN IT UP. secoundly the high RPM is most likley a cause by a bad TPS Throttle Posiction Sensor. and yes the high RPM can cause transmission damage if you do not fix the problem pretty quick .in this case good idea to do soits getting harder and harder to get parts for these cars TPS-25.00 to 45.00 if you fix it your self, simple to do

not really but they do suffer from mass air flow sensors to some extent ,had to renew quite a few on these earlier cars when they get a bit older

There is a cable that goes from the throttle body to the transmission. usually by the throttle body you can adjust the cables lenghth by pushing in the tab. only move this a couple of notches at a time, test drive and see where it shifts. adjust as necessary. 1st gear should be at 15 to 20, 2nd at 30. Let me know if this works for you , we can go from there.
Randy

Does it have a Mass air flow sensor? If yes, unplug it and see how it acts.

Put higher octane fuel in to fill and injector cleaner bottle full. Take car on freeway give it a blast ( don't break speed limit but rev engine maybe in lower gear ) and it will run like new. If problem continues after this then diagnostics at dealership next best option.

on single cam engine its inlet manifold side just behind cam gear rear cover bolted into cam box cover
twin cam its behind cams in middle front of engine in centre of rocker cover remove cover over h.t.leads this one is eazy but tricky

you may have a vacuum leak, check all the vacuum lines around your inlet manifold, carbi etc for cracks or ones that have come off their fitting.
Check all the retaining bolts and nuts around the manifold, they may be a bit loose.