Title kinda says it all.
If I am arrested is there some line drawn that says if I make more than “X” the court is not obliged to provide me an attorney or, even if I am Bill Gates, I can request and get a court appointed attorney? (ignoring if that is a good choice)
Can those court appointed attorneys send me a bill for service later?
(FTR: Just asking…I am not in legal trouble…no need for a fast answer.)
https://www.findlaw.com/criminal/criminal-rights/are-you-entitled-to-a-court-appointed-attorney.html
To determine whether you qualify for a free court-appointed attorney, you may have to gather financial documents and prove to the judge that you lack the funds for a private lawyer. However, some courts may take you at your word (for example, homeless individuals lacking such documentation). Counties may determine eligibility for a public defender in a number of different ways, but your ability to afford a lawyer typically is based on your income and expenses. Some judges may ask you to get estimates from as many as three private attorneys before approving the assignment of a public defender.
You will not be billed.
How about that initial 24 hours when the police pick you up and shine a birght light in your eye in a tiny room? In all the Law & Order episodes, the occasional smart perp says “I want a lawyer”.
IANAL, but my understanding is that if you request a lawyer, the police must stop their interrogation until a lawyer is either hired or appointed. I’m sure different jurisdictions handle determining if the accused is indigent differently, but if the police need to wait until the process is complete, they need to wait.
And I worry they arraign you and send you to Rikers, and …you sit. Because how many of us have a criminal lawyer on retainer?
Great question! I wouldn’t even know how to find one. (I mean, find a useful one, I am aware of the Yellow Pages). I tried to hire a lawyer to help clean up my mother’s estate after she died, and I could barely get anyone to return my calls. I had one sit down meeting with an actual attorney and I think he was drunk.
I can imagine sitting in stir forever, calling different lawyers, trying to read reviews in the jail computer.
You got the bright light, but you forgot the stinky cigar. 
IANAn American lawyer, but that is my understanding of American law.
However, as I recall, @md-2000 is Canadian. And he may be interested to know that in Canada, while you do have the right to call a lawyer, the police do not have to wait until one shows up. They can continue questioning you the minute you hang up the phone, unlike in the United States.
I have my office phone in my home, and I’ve fielded the same question (sometimes at weird hours of the morning): “Can the cops keep questioning me even if you’re not here?” The answer is “Yes,” so my advice is always the same: “Shut the hell up and say nothing until I can get there.”
Canada has no public defenders, but it does have Legal Aid, where you can be represented by Duty Counsel pending the hiring of a lawyer. And if you are not flush enough to afford a lawyer, Legal Aid will appoint one for you, though you will most likely be expected to pick up at least part of the tab, depending on your proven finances.
[End hijack into Canadian law, but it does address the OP’s question in a Canadian context.]
Yes, you will have to do some sort of listing of assets or income to demonstrate your qualify for a court-appointed attorney. Public defenders do not bill you but there may very well be court fees and if they can determine you lied on your asset statements I imagine there are consequences including criminal or financial, but that does not mean the PD is the one who has the responsibility to collect.
In the UK, the lawyer who turns up at the initial interview and tells you to answer “no comment” to all the questions is free unless you call one yourself. The police have to ask you if you want one and cannot interview you if you say you do.
The next stage is an appearance at a magistrates court. If you want to be represented there you will have to pay unless you can claim Legal Aid. To claim that you have to disclose all your income, savings and investments and the bar is very low.
It is crucially important that you don’t ask for a lawyer dog or else your request will be denied.
Yes, no point in expecting a legal beagle.
But I guess that’s the core question - if an American just picked up says “I want a lawyer” or words explicit enough to that effect, do the police find a lawyer for the guy (presumably a public defender) like in Canada, to at least do duty for the 24 hours? Or can they say - “here’s your one phone call, be my guest…”?
Everything I’ve read here and elsewhere says that (at least in the USA) the police will stop interrogating you; so the only question is whether at that point they have enough to charge you?
My deceased first wife was an attorney with a home office. She specialized in banking regulation. Her wry comment:
In banking you meet a nicer class of criminal and they rarely call you from jail at 2am.
She had a point; other than due to time zone differences we never got calls at awkward hours. Awkward moments, sure, but not awkward hours. 
So, if you can afford it, how do you go about finding one from a cell at 2am?
As far as billing goes, some public defenders can bill - here are the fees for the NJ Office of the Public Defender, which can waive the fees if appropriate. If one state allows charges, I’m sure others do as well.
I’ve fortunately never been in such a situation, but I’d think that you call a family member or friend and have them find one for you. Barring that, maybe there are lawyers who find ways to get in contact with you while you are in jail ?
In New York state, at least in my jurisdiction, someone from the public defenders’ office will speak to you shortly after you’ve been arrested to determine whether you’re eligible for legal representation. The vast majority of defendants I see getting arraigned are represented by a public defender at arraignment.
If you are in custody and being questioned and ask for an attorney, someone from the public defenders office will be summoned for you. Most likely they will tell you to stop talking to the police, and frankly not much more advice is needed at that point.
So do they give you your phone to make a call? Nobody knows phone numbers these days.
Most people are not interrogated by the police. They arrest you and that’s that. In cases where they do want to ask you questions, they have to read you your rights (if you’re in custody) and stop questioning if you invoke your right to counsel. But that’s a misleading phrase, what is really is a right to have questioning stopped. They’ll book you or let you go, depending on the state of their evidence. You won’t see a lawyer at that time.
The one twist (in Washington, perhaps elsewhere too) is nighttime DUI arrests. The driver has to make a quick decision to take or not take the breathalyzer. Our courts have ruled that a person as a right to assistance of counsel at that moment, so public defenders are standing by to take the call and give some quick advice. In my day, we rotated that shift (doing it about once a month) and carried pagers for that purpose.
To the OP. Yes, you have to prove you can’t afford an attorney but the proof is generally filling out a short financial forms. It was not at all uncommon for people to get a public defender, but later hire an attorney with money from friends and family
So, if I am arrested and say I want an attorney, if I do not provide my own (because I could afford one), what happens? The state will not give me one and I refuse to hire one. Stalemate (which I suspect I will lose).
As you can imagine, it rarely happens. There’s a public defender at arraignments you can use even if you have financial means. If you don’t hire a lawyer by your next court date, the judge is going to ask you what your plans are. Sometimes the guy says “I’m working on it. blah blah blah.” In those cases the judge says “Come back in a week with a lawyer or be prepared to represent yourself.” They usually came back with a lawyer, or asked for yet more time. Some times they ask the public defender screener to take a closer look at the defendant’s finances. (They tend to err on the side of giving you a lawyer)
I seem to remember one case where the judge said “Fine, you’ll get a public defender, but you’re going to be billed for what it cost the county.” That was one time in all my years hanging out in criminal courts.