1. On the scaled recipe, there is a little purple cross, what does that mean?
2. The scaled recipe is not immediately show up even on the machine from "schall machine scaling" mod. I had to change the mod settings "machines using scaled recipe" from the default into "*". Is this intended? And what does the asterisk mean?

Thank you

On the scaled recipe, there is a little purple cross, what does that mean?

The cross is a multiplication sign, meaning the recipe has a multiplier applied.
Practically, it is just a little indicator to distinguish it from its original recipe. Eye-catching so to let the users know they are picking the correct one.

The scaled recipe is not immediately show up even on the machine from "schall machine scaling" mod. I had to change the mod settings "machines using scaled recipe" from the default into "*". Is this intended? And what does the asterisk mean?

No, totally not intended. The default settings should work perfectly with Schall Machine Scaling.
Either some settings are wrong, some other mods messing with them, or otherwise there is a bug I don't know. I would be interested to know why the default settings did not work for you.

The asterisk * is a commonly used symbol in computer science, as a wildcard character. Here means any length of any characters can fit into this pattern.

Either some settings are wrong, some other mods messing with them, or otherwise there is a bug I don't know. I would be interested to know why the default settings did not work for you.

I did try to restart factorio with only schall machine scaling and recipe scaling, with the default settings. Yet the scaled recipe still did not show up in the assembler crafting menu. Though it is not a problem anymore since I changed the mod setting into an asterisk.

Forget to ask: What machines are you trying to access that?
By default settings, the scaled recipes is supposedly to be used by machines overtier +3 and above. Vanilla machines and +1, +2 machines are NOT supposed to need that, as in principle they are never fast enough to reach the 60 crafting per second limit.

If you cannot see the scaled recipes in machines overtier +3 and above, then it will be a strange case.
Can you show me a screenshot of your mod settings?

It's actually possible to reach the 60 crafts/second limit for some recipes with +1/+2 assembling machines, using beacons. Would it be possible to enable scaled recipes for lower level machines by default?

This would also make the mod easier to understand; like the OP, I was also confused when the mod didn't seem to be working even for scaled assembling machines.

A problem will arise if the default setting would have given access to scaled recipes for +1/+2 machines. Very likely some (new) users will then complain their machines are very slow (like several minutes per production cycle) with the scaled recipes, thinking either or both MS & RS mods being unbalanced and bad. This is what I wanted to avoid.

Therefore, there is no single setting that would be good enough to cover every case. This is exactly why it is provided as mod setting allowing user customization. (Otherwise, I would just have hard-coded it to some "optimal"/"universally good" setting.)

Overbuffing multiple beacons to a machine with fast recipe seems to be... an unusual case to me... I would have achieve the same production speed by replacing such kind of construction with a higher tier machine with much less beacons. Less entities, less material (beacon cost), and probably also less power consumption overall. (Beacons are power-hungry, whether they are working or idle.)
BTW, one of my motivation of creating MS is to reduce beacon spamming. For example, 12 beacons surrounding an AM3 is exactly too-gamey and counter-intuitive to me.

Below is a summary for recipe iron gear wheel (0.5 s crafting time):

This shows using a higher tier machine is better than spamming beacons (in 2×SM3).
The beacons are essential only if having to counter a speed debuff (e.g., 4×PM3 in AM3 giving -60% speed).