How was Howard allowed into space in The Big Bang Theory? He crashed the Mars rover. Even if he did somehow get away with damaging government property worth billions of dollars, how did NASA let him into space? I know he was launched on a Russian based mission, but no space faring country would allow him within 10km of any piece of government owned machinery.
The Government never confirmed or pursued the comment by Sheldon, as they didn't have much evidence to go on. In the episode where he crashed it, S02E08 "The Lizard-Spock Expansion", the three involved end up wiping all the security tapes and fingerprints.
Howard: OK, I guess we have to turn to Plan B.
Raj: What’s Plan B?
Howard: Erase all the hard drives, scrap the surveillance tapes, wipe our fingerprints off every surface and run.
Sheldon: Why wasn’t that Plan A?
Voice from television: REPORTER: A NASA spokesman states that due to the loss of data, they will most likely be unable to determine the cause of the Mars Rover’s malfunction. This is not the first time an exploratory mission to Mars has ended in disappointment.
As for Sheldon blabbing to the FBI in the S04E07 "The Apology Insufficiency", remember how nervous and verbal-diarrhea like he was? Between Sheldon being a fairly unreliable witness (what prosecutor would put him on the stand?), the lack of physical evidence or collaborating statements, and Howard being a prominent aerospace engineer with a highly respected career (Sheldon's derision not withstanding) they have very little chance of winning.
Howard goes to the ISS because his telescope design was chosen, and they needed someone familiar with the design to install it. Considering the politics (re: budget reasons) behind scraping the plan due to some unfounded allegations, NASA/the Government thought it was worth the risk.
That said, an astronaut typically requires a top-secret security clearance as well, so this is just another example of the show giving rule-of-funny precedence over 100% strict realism. Don't let facts get in the way of the plot.