Picking out correct MOSFET for solenoid PWM control?
Let's first rule out static losses as the cause for your troubles: Your MOSFETs have an on-resistance of approx. 100&nbsp;m\$\Omega\$ (or something much lower). With a load current of not more than 4&nbsp;A, the power dissipation for a full (100&nbsp;%) duty cycle should not be more than

PV,&nbsp;max&nbsp;=&nbsp;RDS,&nbsp;on&nbsp;\$\cdot\$&nbsp;I2&nbsp;

PV,&nbsp;max&nbsp;=&nbsp;100&nbsp;m\$\Omega\$&nbsp;\$\cdot\$&nbsp;(4&nbsp;A)2&nbsp;

PV,&nbsp;max&nbsp;=&nbsp;1.6&nbsp;W

To adress your question #1: Don't try to use a MOSFET with a super low RDS,&nbsp;on  when you don't have to. The low on-resistance comes with the price of a larger gate charge, making it harder for your MOSFET driver to switch it fast. Also, a DPAK should be able to handle the static losses with a PCB like yours (your question #2).

Having checked this, and reading your note on not being able to use more than 40&nbsp;Hz as a PWM frequency, I suspect something is wrong about getting a clean signal from your µC board to the power PCB (question #3). It could happen that every time you switch on the MOSFET, the ground voltages of your power circuit and your small-signal circuit bounce with regard to each other, causing your MOSFET to switch quite a number of times whenever it should just switch once. How long is the connection between the microcontroller and the MOSFET driver's input? How does the overall supply wiring look?

Edit: Now that things are a bit clearer after you have added your schematic, I feel that your input side (driver IC and MOSFET gate) is in danger. The flyback energy released by the solenoid after switching off needs a place to go. Your paralleled 1&nbsp;µF and 100&nbsp;nF capacitors may not be enough, and the voltage may rise beyond the max. voltage allowed as VDD for the IC or as  VGS for the MOSFET. It is not clear how long the wire from the next stiff source (read: good capacitor) to your board's input is, and I strongly recommend a large, local electrolytic capacitor (1000&nbsp;µF, 35&nbsp;V).