Shouldn't I use a mixer/pre-amp and connect the output to the 3.5mm jack on my laptop? Would I severely compromise audio quality while doing this?
I am on a budget. Can I get 48kHz 24bit audio using the built-in sound card?
I use a dynamic mic.
This depends on the laptop of course. Many computers have low-budget audio circuits. Sometimes they're badly shielded from the rest of the computer so you get noise. An external interface has better shielding, and usually has higher-performance audio hardware.
I don't think 48 kHz 24 bit sampling is common on built-in audio hardware.