Satellites can and do use HF for communications.
The first example would of course be Sputnik, which transmitted at 20 and 40 MHz.
Amateurs use HF to communicate with satellites. According to Amsat,
mode A: This mode requires a 2 meter SSB/CW transmitter and a 10 meter SSB/CW receiver...
Mode K; This mode requires a 15 meter SSB/CW transmitter and a 10 meter SSB/CW receiver... This mode is unique in that it can be done with a simple HF rig.
HF has advantages in its smaller Doppler shift, and reduced path loss, allowing omnidirectional antennas to be used more easily.
For the physics part of your question: the reflection from a layer of the ionosphere depends on electron density, frequency and angle of incidence (which changes its effective thickness).
At normal incidence, frequencies of below about 7-10 MHz are reflected, higher frequencies pass through. See Wikipedia for a graph of reflection against height.
This page gives a great overview; (written with communication in mind, but what isn't reflected is transmitted). The equation for MUF is F_normal / ( sec θ ). So it's only for very long paths, grazing reflection, that the ionosphere ever reflects a 28 MHz signal. It would usually pass straight through the ionosphere, down to a fairly shallow angle.
Your last question makes it sound like you hope to use HF from a cubesat when it's over the horizon, not in line of sight. This would be highly unusual and would certainly not help with your data rate problem.
- If you choose a frequency which easily penetrates the ionosphere, then it won't bounce around inside, to get to you, it'll just reflect out again.
- Cubesats have very limited power, HF requires plenty of power to communicate around the world, 10s or 100s of watts.
- If you're trying to send more data during a pass, then the right solution is to go higher in frequency, where antenna gains are higher, there's less atmospheric noise, and more bandwidth available. HF modems run at 1200 bps, WiFi runs at 11 Mbps and more.
That all said, there is one tiny balloon project, launched to fly around the southern hemisphere, that uses HF to phone home. It's only 30 km high, but it transmits just a few mW using wspr, and can be heard around the world, with luck. Bear in mind that it sends nothing but its position and callsign.