Q2: Second, I don't really need funding. I'm a working professional and I wouldn't need a stipend. Should I indicate this in the email?
Two things here. Are you contemplating part-time. Your "I'm a working professional" would be construed as seeking PT.
There's no rule, AFAIK, that stipulate indicating outright from the onset.
Nonetheless, often, if a potential supervisor is interested in you, (s)he/they will set up a short session with you. Both parties will get to engage on interest and key areas. Issues of stipend can be engaged on here.
Be it as it may, whether you indicate in your 'introductory' email or not, keep it succinct and polite as you can.
Q1: ... Something like:
I am looking to do a PhD in field X. While I have ideas about a research proposal, I would value your guidance in producing the final proposal.
In my view, there's no clear cut approach.
You may consider along the line
I'm looking at PhD in field X ...
I have looked into subfield X.1 ...
I have researched into subfield X.1
Your work in subfield X.1 caught my interest and I'm keen engaging further on subfield X.1, X.1.1, X.1.2 ...
In essence, simply mentioning I've ideas in X without a 1 or 2 sentence of what that/those idea(s) is/are might not suffice. Of course, you are not compelled on specifics: you can be broad (within field X).
Just to mention:
In the UK, some universities list
- studentship (with awards/stipend)
- open PhD research projects (funded or unfunded)
- units/centres research areas/interest
PS: Each has its own dynamics within the interested candidate and potential supervisor/advisor.
In certain instances, not all, for competitive studentship, the dept/doctoral school/college might request you submit a formal application. Post interview and upon acceptance, the proposal writing process takes place with (possible) supervisor/advisor.
One writes and submits around the studentship's research topic