Can I sue a dealership for leasing me a car with a different mileage from the car history report?

The car history report is not a legally enforceable document: it is evidence of what the mileage should be, but nothing more. If you believe the dealer has falsified the mileage, you could sue the dealer for fraud; you would have to prove to the court, by a "preponderance of the evidence" (that it is more likely than not) that the dealer kinowingly or intentionally falsified the mileage. Note that the car history report may not be enough, by itself, to prove this, in the face of contrary evidence (e.g. what it says on the odometer) and also if the car came to the dealer with the mileage already falsified, the dealer would not be responsible for the actions of prior owners.

As to the transmission: again, if you if you can show that the dealer knew of the transmission problem and hid that from you, the dealer may have committed fraud and you may have a basis to seek monetary compensation or rescind the transaction. However, if the problem developed after you took the car, or if the dealer did know of it prior to leasing the car to you, the dealer would not be liable. You would, in that case, only be able to enforce any rights you have in any warranties.