Will opting for a balance transfer card increase rate on future credit cards?

When you take out a card, the details are there in the credit report.
When you're shopping for a card, it may reduce the score momentarily. Your credit utilization ratio will also increase in one card. But, at the same time, your credit limit will also increase since you've taken out a new card with a definite credit limit.
And, when you use the card responsibly, it will help you increase your score.

Yes, it will be there on your credit report. But once you start repaying your credit card bills in full an within time, your score will increase with time.

No they won't increase the interest rate on the credit card as long as you pay the full balance within the introductory period. See, when you take out a balance transfer card, you pay zero percent interest on the balance. If you can't pay off the balance within that time, you're doomed. The interest rate will increase all of a sudden and it will be difficult to pay off the debt. Your credit score will drop if you fail to pay off the debt. Potential creditors will increase interest rate in that case.