First, if you want to read about the medieval peasantry, start with Paul Freedman's Images of the Medieval Peasant, which is actually one of the more enjoyable monographs on medieval history I've ever read.

The easiest thing to say here is that "slavery with white people" is a totally incorrect way of describing the medieval peasantry. There were in fact actual white slaves in Europe for quite some time, and they were legally different from peasants/serfs (although they disappeared in England during the century or so following the Norman Conquest). In terms of the two systems, there are so many points of difference that it's really not fruitful to compare the two. The main points of difference would be legal status (serfs weren't slaves: this has many implications), living conditions, work performed, family life and marriage, the geographic differences between England and the American South (ploughing a barley field and feeding your own chickens in a drizzle is very different from being worked to death on a tobacco plantation in 100 degree weather with 90% humidity), etc. So, basically everything.

To just examine the legal aspect: Explaining the legal statuses of various peasants (free v. unfree [unfree != slave], degrees of freedom, etc.) is extraordinarily complicated. You often hear that "peasants were tied to their land", which is true, and that they owed various things to their lord. The actual circumstances of this was extremely variable sometimes literally person to person. A totally free peasant only owed their lord basic taxes and tithe to the church. A totally unfree peasant owed a massive amount of taxes (often "in-kind", meaning say that they had to give the lord a chicken every year or a bushel of apples or whatever) and services, such that he would spend much of the year laboring on behalf of his lord, either directly or indirectly to pay the taxes. Even so, an unfree peasant lived in what was more or less his own house (even if he didn't literally own it -- his lord might not literally own it either!) and could do things like raise extra chickens or what have you for himself. He wasn't in the slave quarters barely eating and totally unable to better his position, the situation most slaves in the Americas found themselves in. Peasants who fell in between free and unfree might owe their lord fewer days in service and less in taxes and thus have more time and resources to use for themselves. There were more issues than just taxes and services, unfree peasants could owe their lord other obligations like having to pay a fee to get married (technically, as you hint, there were regulations on who could marry whom. But these rules were often circumvented). But that's the gist of things.

There were also other legal issues. Peasants had basic legal rights and were brought to trial with juries (in England) and depending on status had access to different types of courts, were allowed to marry, had full rights to their children (ie, nobody was shipping their children across England for profit), etc. Although there were certainly abusive lords during the Middle Ages, peasant women weren't supposed to be raped, peasants weren't supposed to be whipped and beaten, certainly not without cause, they weren't supposed to be unlawfully stripped of the land they worked, etc. All of these things are totally the opposite for slaves in the US, who had no legal rights, couldn't legally marry, their children were the property of their masters, etc.

As to your specific questions:

Were serfs beaten with the same frequency as slaves?

No. They mostly weren't working in a system where an overseer was forcing them to do their work, so there was no need for frequent beatings.

Were serfs really allowed to become free if they escaped their lord for "a year and a day"?

Yes. "Stadtluft macht frei", as they say in German, or city air makes you free. This was customary, and abolished in the thirteenth century in the Empire, but plenty of serfs across Europe fled to towns and cities in order to escape their original lot.

Did the rape of serfs happen with the same frequency of the rape of slaves?

I wonder what source you are quoting? Lords weren't supposed to go around raping their female peasants, but some did. I don't know that we know enough about the rates in either circumstance to definitively compare them.

was there a house serf/field serf dichotomy?

No, at least not in the way there was in the American South.

How regularly were serfs manumitted?

They weren't slaves, therefore they weren't manumitted.

Were English serfs allowed to be educated?

Yes. There weren't a lot of people running around doing educating, though. Usually in order to receive an education they would need to join a monastery, which was not common for peasants. But it did sometimes happen. In cities, it was more common for commoners to become educated in order to become lawyers or bureaucrats.

Flemish sources from this period report that marriage between free men and slave women was restricted. Is this true of England as well?

Yes, although as I mentioned this was oftentimes not followed. I'm not sure if you rendered unfree into slave, or if it says slave in the original? Technically, there were all sorts of rules for who could marry whom (ie, unfree v. free v. slave) and how that affected their free status and that of their children (check the Salic Law for a very early example of these rules). A free man had a lot of incentive to not marry a slave woman because it would endanger his own status. On top of this, by the time you're talking about, slavery had more or less disappeared in England, so there wouldn't be that opportunity.