Great Question!
To start with we should note that John the Baptist, Jesus and the early church all preached repentance(See Mark 1:4, 14-15; Acts 2:38). Paul descibed the gospel he preached this way in Acts 26:20 "...I preached that they should repent and turn to God and prove their repentance by their deeds." We have to conclude from Scripture that repentance is a critical part of the gospel's message.
As we are saved by grace through faith, some may say repentance need not precede our coming to Christ and being saved. I personally have trouble understanding how one could come in faith to God without repentance. If one worships "god as they want him to be," they may be so bold as to approach him unrepentant of sin. But that god does not exist and cannot save. The God who is and was and is to come as revealed in Scripture is loving and forgiving, but also just and holy. When one comes to God as he is, I expect they understand that they cannot stay as they are. God alone has made salvation possible, but for us to receive that blessed gift, we must by faith turn from sin and toward he who saves.
Concerning the passages mentioned: With the sinful woman, as one noted earlier, Jesus clearly expected repentance. Also note that Jesus told Nicodemus in John 3:17 that "God did not send his Son into the world to condemn the world, but to save the world through him." Why? Because those who do not believe are condemned already. Based on these things, I do not think we should be bothered by the passage of the woman caught in adultery. Whether her encounter with Christ birthed belief in her heart or not, we know that he expected her to walk in repentance.
The other account mentioned is very interesting. Jesus seemed intent not only to show that he could heal, but also forgive sin. Sometimes sickness is caused by sin (John 5:14). Perhaps that is why Jesus said what he did. However, I think it is reasonable to assume the paralytic also had faith.
I am wary of assuming that both of these people received salvation as we think of it based upon Jesus' response to them. It is possible, but we should meditate on it before we assume that as the reality.
I can't say I understand it all, but in light of the rest of Scripture, I continue to conclude that one cannot receive salvation without turning from sin.