No, you cannot avoid losing HP on a regular attack. You can gain HP by killing red enemies and using the proper special attacks, but I suppose if you played the tutorial (hint hint) you would/should know all of this.
You aren't getting your badges right away because this website is known for not being the most responsive website in the history of human life. If you are getting a glitch/temporary lag-up, re-get the badges and they should be awarded to you.
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