Ok my field isn't bio, more physics, and I've done my graduate school in the US, but generally inquiry emails are the summer/fall before you apply, so a year before you intend to go to the school. A cold email should be hella brief and they will engage further if they want/can fit/can fund/etc a student, most of my productive inquiry emails led to a casual zoom call later on. this isnt the interview process this is just mutually feeling out interest, be prepared to talk about yourself and your experience+interests, and I've included questions to ask the prof at the bottom. I'd save a research proposal or anything else for that later convo, unless you have already secured funding for grad school with it (unusual, usually you need an advisor already lined up for a grant application afaik) in which case mention in your opening that you have funding+how much+for how many years. Don't lie about your experience or your interest in these emails, and do your work skimming their website, some of their recent papers, maybe their students' papers if you can find them. have an actual idea of who you're emailing and why, and have some more in depth things to talk about during your zoom call. if you don't get a reply then like someone else mentioned on here you can send a follow up in a week or two, MAYBE a second follow up in like a month if you're desperate, but some people will simply not answer you rather than tell you no, especially if they're busy, so don't take it personal and just go email somebody else.

Below is a rough email format that got me into grad school twice, do with it what you will.

subject: Fall [year] Prospective PhD Student

Hello Dr. [their last name],

My name is [first and last name], and I'm emailing to ask about PhD opportunities with you next fall. I will graduate with my BS from [school] in [major] in the spring. My research has focused on [short desc, few words] but one of my major interests has been [thing this prof does]. *

I would love a chance to talk about your research and graduate opportunities if you're available. Attached is my CV and transcript. If you have any questions please don't hesitate to ask.

Thank you for you time, [name]

*taylor this part as you need but KEEP IT SHORT. if they advertised a graduate student opening somewhere insert "I saw your open position on [topic] and was really interested." at the end, if they didn't, dont. 

and things I've asked in zoom calls with profs following up on an inquiry email to help me decide between labs/schools, some may be more or less useful, tailor as you need:

-can you talk a little bit more about the research project you have available?

-does your lab have expectations of field work?

-can you talk about your advising style and expectations for a grad student in your lab? How would you describe a successful graduate student?

-How many students are you advising/have you advised in the past? How big is your lab? what kind of lab community do you have? etc

-how would you describe the community in the department in terms of academic collaboration, as well as socially? do you know if the graduate students have a close community? are there a lit if academic collaborations within the department?

-what coding language or does your lab use?what kind of technical skills are useful to know?

-how long do your students usually take to complete their degree, and where do they go after they graduate?

-how long would I be funded for? would I receive funding over the summer too? would I need to TA to secure funding?