If I make dua during tahajjud without praying salah, does my dua still fall under the Hadith where Allah mentions "Is there anyone to ask Me, so that I may grant him his request?" Would it be a greater chance of getting accepted if I had prayed salah?

Ustadh AbdulAziz al-Haqqan

Yes, the hadith is general and is not restricted to salah, so your dua in that time is included. Combining it with tahajjud is the higher level, and the sujud of your salah is itself a place where dua is most likely answered.