Hello @JustAnotherPoster00. Sorry for the late reply, I only just noticed your message! And thank you for sharing those quotes. Actually, I think you've misunderstood them slightly, so I'll share the Christian view of these quotes. 
“For truly, I say to you, till heaven and earth pass away, not an iota, not a dot, will pass the law until all is accomplished. Whoever then relaxes one of the least of these commandments and teaches men so, shall be called least in the kingdom of heaven; but he who does them and teaches them shall be called great in the kingdom of heaven.” — Matthew 5:18-19
The accomplisment here refers to the death and resurrection of Christ. When Christ died, His final words were "It is finished."
“Did not Moses give you the law, and yet none of you keepeth the law” (John7:19)
If we read this verse, we can see that this was said by Jesus to reveal the hypocrisy of those Jews who wanted to kill Jesus and tried to use the law of Moses to justify this. Obviously it was also said prior to His crucifixion.
“It is easier for Heaven and Earth to pass away than for the smallest part of the letter of the law to become invalid.” (Luke 16:17)
In the version you sent, the word 'invalid' is used, but in other versions we can see 'fail', 'become void', 'be overturned'.
In Luke 22:20, Jesus introduces the new covenant, which was ratified by His blood. ('And likewise the cup after they had eaten, saying, “This cup that is poured out for you is the new covenant in my blood".'). But this new covenant does not make the law invalid, as the verse you sent says. It doesn't cause it to fail, become void, or be overturned. Nothing like this. The new covenant, instead, COMPLETES the law. The law is not nullified by Christ, but it is fulfilled by Him.
So this is why, as Christians, we no longer need to follow the laws given to the Israelites through Moses 