As I understand it, the Welsh marches were ruled by Norman lords who basically seized control by themselves. They swore fealty to the King of England, but were not actually part of the Kingdom of England. Is that correct? How much control did he have over them, and did they use English law, Welsh law, or their own laws? How much power did they have over their subjects?
The Welsh marches and their Earls are quite an interesting part of the Anglo-Norman Kingdom, so forgive me if this post doesn't become too in-depth!
When William I succeeded in his conquest of England he appointed three of his most trusted and capable followers as Earls, William FitzOsbern as Earl of Hereford, Roger de Montgomerie as Earl of Shrewsbury and Hugh d'Avranches as Earl of Chester. They were certainly not usurpers, but they were granted special privileges which continued to be recognised throughout the 11th and 12th centuries (and beyond). They were most certainly a part of the Kingdom of England, but the privileges they enjoyed were quite significant, and this may have led to the confusion expressed in the title question.
The most significant privilege was perhaps the ability to ignore any peace or treaty the King had made with the Welsh Princes. For example, if King Henry II were to make peace with the Prince of Gwnyedd, this peace treaty would not apply to the Marcher Lords and a separate peace would need to be negotiated with them if the Ruler of Gwynedd wanted 'total' peace with his Anglo-Norman neighbours.
Other privileges included being exempt from royal taxation and being able to create their own markets and other economic assets. These differences are primarily due to the Marches being a 'frontier' zone, and as such were more likely to be raided and less security/profit was to be had then say central East-Anglia. However, this also meant the Marcher Lords were free to expand at the expense of their Welsh counterparts, so the very instability which made the marcher territories unstable also made them more profitable. Ultimately they were still vassals to the King and did have to support the King during times of war.
They encouraged migrants from England, Normandy, Brittany and modern day Belgium to settle in the areas of Wales they controlled. These settlers were often attracted by economic incentives, such as cheap land or favourable taxes. It can be argued that this gave the Marcher Lords less power over their subjects, or at least the migrants, but they did also enjoy the right to levy men into their forces as they saw fit, like other English Earls, so this is debatable.
Your question about law is tricky, as often the marcher lords would differentiate between which laws they followed, sometimes following the English laws and sometimes following the Welsh. This resulted in several debates as to which laws were to be followed in different cases.
In sum, although the Marcher Lords were part of the English Kingdom, they had privileges which other English Lords did not. Their control over their subjects was certainly no less then other English Lords, unless it was impeded by terms they themselves offered to attract new settlers. The English laws they followed were often exchanged for the Welsh and vice versa.
I hope this answered your question somewhat!
Sources: Many of the rights are renewed in the Magna Carta, article 56 is of particular interest.
A book which discusses the status of the Marcher Lords, although the work also deals with a great many other Kings, Princes and Lords, is J. Benham's 'Peacemaking in the Middle Ages', and may be of interest if you'd like to read further of Medieval diplomatic practice.