Regulate, convert to DC and store current from home made AC generator to charge laptop
This would be an interesting project to learn some electronics but it may not be a practical one. For example, you don't say how you will power your generator.

First understand that your laptop power supply's relationship between power in and power out is given by 

Power in = power out + losses


If you have a good power supply the losses will be about 10 - 20%.

You should also know that \$ P = VI \$ so that your PSU can output \$ P = 19 \times 4.5 = 85\; W\$. If it were 85% efficient it would draw 100 W from the mains. At 100 V this would be 1 A. At 200 V it would be 0.5 A. The 1.6 A figure is worst case surge on power up.


  1a. First, I need to decide if I should make my generator output at least 100V and provide at least 1.6 A to the laptop's charger then let the charger take care of the battery.


If I had to do it this is the way I would do it. The PSU regulator offers some protection to the laptop.


  1b. But if the generator creates 19 V and 4.5 A and sends this directly to the laptop, less work is necessary to rotate it, I guess.


It would, in theory, be 15% less because you wouldn't have the losses of the PSU. This would probably be replaced by the losses in your new circuit.


  1c. Why would I produce high voltage then have the charger reduce it?


Because your mixer windings are rated for \$ I = \frac {P}{V} = \frac {300}{230} = 1.3 \; A\$. They won't supply 4.5 A. (I have assumed you live in 230 V territory. You didn't add that information to your post or your profile.)


  
  I need to know: how to regulate the current, how to transform the voltage as needed, how to store it temporarily, at least to keep it steady.
  


Yes you do. And you will need to research this rather than hope someone here is going to design it for you.


  [Should I] convert the AC current to DC using a diode bridge, and add a smoothing capacitor.


If you want DC or a switched mode power supply this is essential unless you are generating DC.


  Regulate it with a Zener diode (and required resistors to limit current flow) or a regulator (preferably to 12 V).


This would be a hopeless way of regulating. It is not used in commercial designs as it is inefficient and regulation is poor.


  Convert the voltage from the regulated one to 19 V then use a capacitor to keep the amount if energy steady and provide it to the charger / battery?


You need to study SMPS power supply design. Your question implies that you don't understand enough to start this project yet.

Cautions

The risk of injury are high if you use voltages over 100 V.

You have to factor in the risk of destroying a laptop if you get something wrong.

I suggest you use the generator to power some LED strips instead. If they blow you've lost €10.