Yes, by filing a PCRA petition. To be successful you must show that your lawyer was constitutionally ineffective, and that his/her futility resulted in your conviction.
The Commonwealth (district attorney/attorney general) has the right to automatically appeal pre-trial orders granting a defendant relief, if he/she can show that his/her ability to prosecute the case is substantially impaired.
A prosecutor can also file appeal challenging the legality of a sentence.
Yes, you have the right to represent yourself on appeal. In fact, filing a notice of appeal is not overly complicated. However, identifying, preserving, and arguing issues on appeal is extremely complex. While pro se litigants have won noteworthy appeals, it is typically an uphill battle.
It depends, but it is typically a long process (a year to many years) unless the trial court grants the relief requested in a post-sentence motion.
Yes, however, the right appeal is limited to review of: