Spanwise flow and wingtip vortices in forward-swept wings
There is no magical switch that gets flipped once you cross a specific sweep angle. Things are gradually changing with the cosine of the local sweep angle. Now let's look at your questions one by one:


  What happens to the wingtip wake on a forward-swept wing?


Nothing special. It is part of the full wake system and behaves like in any other aircraft of comparable aspect ratio and lift distribution over span. For high sweep angles and high lift, the wing torsion will increase local loading towards the tips, making the tip vortex relatively stronger.


  Does sweeping wings forward completely eliminate it, and if it depends on sweep angle?


Why should it? No, the tip wake of the wing with more forward sweep will be stronger for comparable wings with different sweep, for aeroelastic reasons as explained above.


  How to estimate sweep angle, at which the spanwise flow starts to flow to the root?


See here for the reasons for spanwise flow. As soon as forward sweep results in a lateral local speed component, spanwise flow will happen.


  Considering a tapered wing with only trailing edges swept forward, is it enough to get wing characteristics (primarily spanwise flow) like (or close to) forward-swept one?


With a straight leading edge, flow will not exhibit any sweep effect. With a large taper ratio, forward sweep can still become significant towards the trailing edge, but most flow acceleration happens around the nose, so overall sweep effects will still be small.