Can I use the modern looking shifters with the older model derailleurs without any problems in shifting?
The general rule is that all Shimano road components of the same 'speed' (8, 9, 10 etc) are compatible. There are exceptions though, most people seem to not realize that Shimano publishes very detailed specifications and compatibility information which is worth checking.
Check Rear Drivetrain Compatibility, search on page for 'RD-24400', you see there is a connection for both SS and GS types to the ST-R2000 shifter.
Check Front Drivetrain Compatibility, search on page for 'FD-2400', you see there is not a connection ST-R2000 shifter. The R2000 shifter is only compatible with the FD-R2000 front derailleur.
It's generally known that Shimano used the same 'classic' cable pull ratio for all road groups up to 10 speed which is responsible for wide compatibility, but that holds most true for rear derailleurs. I suspect that Shimano changed the front cable pull ratio on the R2000 groupset for better front shifting.
You could try the older front derailleur with the newer shifter and it may work OK. If it doesn't swap out the front derailleur, they are relatively inexpensive compared to the shifters.