Running both a 120v and 240v extension cord in generator simultaneously
Yes and no. It not as simple as keeping the wattage under the maximum value.
The load needs to also be balanced if you have 120v devices the load needs to be on both legs equally or close.  220/240v loads are usually balanced but not always. (A dryer is a example of an unbalanced 240 load).
1 120v leg you can normally pull 2400w beyond this will probably trip the protective circuits. (For the single 120v outlets)  splitting the and putting the load evenly on each leg it will work so to not overload the generator.
you may be able put ~6000 w on the 220/240 but then it depends how the 120v circuits are loaded 1500w on each leg will work but a large imbalance will overload the generator even if the total is less than the generator rating.
I mention the loading because years back a man set up his own system on the first power outage he kept tripping the generator over temp safeties (ruined the generator) they would not warranty it and called it abuse, I was referred to set up the new one.
His problem 80/30 he had 30% on the 240 load then 50% on L1 leg nothing on L2 even though he had less than the total load it was so heavily imbalanced it smoked the coils from being reset.
I went out and he was really upset he had done an excellent job setting it up but did not understand balancing and the inspector did not check it when he signed off the job.
I moved 3 circuits over to the other side and then asked him what he would add if he had power.  I added 4 more circuits and his total was still well under the rating but well balanced.
I explained how to check the loads with the meter on the generator and to make sure L1 was close to L2 and he would be fine with 1-2 below the max but the large imbalance and resetting was what damaged the original generator.
He did not believe me and turned off his main and ran the generator for close to 250 gallons of propane without a problem.
So as the question is asked yes or no can be the correct answer depending on balancing of the 120v loads.