Re: Religious Restrictions on Training
2:222 is about a husband having intercourse with a menstruating wife. Don't do it because its not hygienic.
4:43 and 5:6 applies to not being able to Pray after amongst others having had sexual intercourse (this applies to both man and woman) until you have bath (hadath) to purify yourself.
So I don't see a valid argument about discrimination to women or not being able to touch women in practice. We certainly don't believe women have cooties. And here's a religion that frequently espouses that the door to heaven is under the mothers feet. When asked who is more noble, the Prophet answered 'your mother' thrice before 'your father'.
At the end of it all, some people believe in the material and the base value. Others have faith that God knows our true intentions. If you believe that in practice, touching women will illicit sexual connotations, then you are wise to prohibit that hidden desire. But if your intentions are noble and it hasn't cross your mind, why shouldn't you train with the opposite gender.