1. Is there any downside to converting my rollover IRA to a Roth?
The big downside is that your taxable income will go up for this year. If you don't owe any taxes now, you may after converting your IRA. For maximum tax efficiency, you may want to calculate roughly how much more you could earn without owing taxes and only convert that much this year. Do this bit by bit until all your savings are Rothified.
2. I have Vanguard flagship service through my grandfather. Can they do this for me and walk me through it?
Vanguard can definitely help you with this. Whether the flagship service representatives would do it specifically, I can't say.
3. If I transferred my rollover IRA to a Roth, would I still be able to fund my existing Roth? Would they be separate accounts or one account?
Yes, conversions don't affect your ability to make new Roth contributions. You should be able to put it all in one account.
4. Is there any downside to converting Mr. FP's 403(b) to a Roth IRA?
Same answer as (1) above. You'll have more taxable income this year if you make the conversion, so plan accordingly.