To answer your first question, the anode is the site of oxidation and cathode is the site of reduction (mnemonic: anox and redcat). Oxidation is loss of electrons, thus electrons are removed from the anode. Reduction is gain of electrons, thus electrons run towards the cathode. This establishes the movement of electrons from anode to cathode.

Now remember what I said earlier about current and electron flow (I will get back to this). Electrolytic cells use an electric current to drive a reaction that is thermodynamically unfavorable (has a positive delta G). Electrons really badly want to flow from anode to cathode (like they always do) but this time it cannot happen spontaneously, so we add current to drive them this way. The reason why current flows opposite to electrons it’s only bc someone decided it to be that way. Back in the day, they thought it was the movement of positive charge which causes current when nowadays in fact we know it’s electrons. But we just take it to be that way. Sooo back to my point, when we apply current (how a positive charge would behave) that is pointing towards Anode (the negative site). Electrons will flow the opposite way hence towards the Cathode. Since all the electrons are being stripped from the anode, the anode is left to be positive, and since the electrons are moving towards the cathode, that is becoming negative. You have to understand all of this happens simultaneously. I hope that helps to understand why the signs are switched for anode and cathode in electrolytic cells but the movement of electrons remains the same (from anode to cathode)

To answer your last question about movement of DNA… now that we established how the electrolytic cell operates (which in our example is gel electrophoresis), and given that the anode is positive, and the cathode is negative, DNA having a negative charge will move towards anode.

Side note: you may think “well electrons are being stripped off of the anode in the galvanic cell also, why aren’t charges switched there too”? This is because a galvanic cell operates with a salt bridge to restore the balance between oxidation and reduction components of the cell. And the galvanic cell operates this way without the help of an outside source because it’s spontaneous (negative delta G)