There is no solid rule on how far back they can go but a judge would make a decision before trial. Yes you do have the right to speedy trial and suppression hearings prior to trial. You should be represented by counsel and discussing these matters with him/her
Ten years from judgment or release from confinedment, which ever is later. FRE 609(b). You always have a right to a speedy trial. Suppress what? Evidence is often hard to suppress. You might get a hearing, but you don't provide any information on whether it would be granted. You presumably have a lawyer, what does your lawyer say?
Although 10 years is typically how far back they can go to bring up previous convictions, it depends on the nature of the previous conviction and what they are going to use it for. If you testify, then the prosecutor may want to use the previous convictions to attack your credibility. If not, it needs to relate in some way to an issue in the case. For example, a previous drug conviction could be used to prove previous knowledge of drugs to rebut a claim, for example, that you didn't realize there were drugs in the baggie. You should discuss these issues with your lawyer.
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