The wheel has 32 spokes total, made of steel
Good! Although for even better results, 36 spokes would be optimal for 700C (622mm bead seat diameter) wheels. However, using 36 spokes would require new hub and rim.
I hope it's stainless steel. If not, you have an opportunity to replace the spokes with stainless ones.
each spoke has a width of 2mm across all its length.
If you decide to replace the spokes, replace them with 2.34mm - 1.8mm - 2.0mm triple butted spokes such as DT Swiss Alpine III (note it has to be Alpine III as the regular Alpine is 2.34mm in midsection which is a really poor choice).
The idea is that the 2.34mm bend is very strong so the spokes do not break there and the 2.34mm bend can be supported at aluminum hubs in a better manner. The 1.8mm midsection is elastic, distributing the rider's weight over a larger number of spokes due to the elasticity. The 2.0mm threads are compatible with regular nipples.
At the same time, use DT Swiss Pro Head nipples. They have a spherical seat as opposed to a conical seat, better supporting the load of spokes that are at an angle.
I weigh 68kg (150 pounds).
For you, 32 spokes might be barely enough if the wheel is well built. For me (110 kg), I really do need 36 spokes.
I don't have a tension meter, but it seemed pretty tensioned,
If you plan to replace the spokes, a tension meter seems like a good idea, but do note that not all tensiometers come with a tension table that is valid for Alpine III spokes. I usually set the tensiometer reading to halfway between what is optimal for 2.0mm midsection spokes and 1.8mm midsection spokes. The tensiometer reading is a number that has no meaning without the table. So I essentially look at the table for 1200 N to see what the reading would be for 2.0mm spokes (example: 45) and for 1.8mm spokes (example: 55) and then use the reading halfway between those (in this example: 50). I understand that a DT Swiss tensiometer would come with a table valid for Alpine III as well, but might cost a lot.
I'm not sure whether the spokes are bad quality, the hub is, or both. The spokes seem to have bitten into / enlarged the holes in the aluminium hub, and I don't know if that's normal.
It's entirely normal. To reduce this effect, you need to better match the spokes to the hub holes. The hub holes are usually 2.6mm. A 2.0mm spoke is a really poor fit. The 2.34mm / 1.8mm / 2.0mm Alpine III spoke is a much better fit.
These spoke holes are by the way the reason 1.8mm / 1.6mm / 1.8mm spokes are not recommended even though they would be clearly superior to all other spokes when only considering their elasticity. The 1.8mm / 1.6mm / 1.8mm spokes would require a very small spoke hole in the hub, so small that installing a 2.0mm (at its bend) spoke to the hole would be hard. Because we don't have these small spoke holes in the hub, the superior choice (1.8mm / 1.6mm / 1.8mm) has to be replaced by 2.34mm / 1.8mm / 2.0mm.
Is it fine to replace all the spokes, or is the hub faulty and is better to buy an entire new wheel?
I think it's doable to replace the spokes. Because the spokes have modified the spoke holes in the aluminum hub, be sure to install the new spokes in exactly the same orientation as the old spokes. Otherwise the hub will be weakened even more.
And before you do anything, purchase "The Bicycle Wheel" by Jobst Brandt. You need to properly stress relieve the spokes in the wheel to get a durable wheel.