While 1 would usually be premature before a colorable offer were made, it is not unethical. It does show a marked aversion to taking the case to trial, though, and mitigates in favor of retaining a trial lawyer rather than a general practitioner.
2) There is nothing wrong with asking the client for written authority to settle a case. It is your use of the word "insist" that gives me pause. How did the lawyer do this "insisting?"
3) while I don't know about Ohio, New York does not have a loser pays statute. Costs are borne by the respective parties, though sometimes a bill of costs will be awarded by an appellate court. This does not include things like fees. Loser pays statutes disenfranchise those of lesser means and serve to chill access to justice. But that's just one lawyer's opinion.
I am a co-author of WEITZ ON AUTOMOBILE LITIGATION: THE NO FAULT HANDBOOK. The opinions expressed in this answer are not intended to be taken as legal advice. These opinions are based on New York practice. I may be contacted at 212-553-9300.
There is nothing unethical here. Medical negligence cases are risky and expensive. Just because you have experts to say there was negligence does not mean the jury will decide in your favor. THe insurance industry has poisoned the jury pool. I am sure you believe that there are too many frivolous lawsuits filed and that juries award money in cases that should never have been filed. You have bought into this fiction. It is simply a lie!! What your lawyer is doing is making sure you understand that going to trial is a gamble. He/she simply does not want to be in a position where you reject a settlement and if you lose, you blame him/her for not settling the case. Do you really see a problem with this? Even if your lawyer recommends you reject a settlement, the final decision is yours and he/she again does not want you later to blame the lawyer for not insisting you take the settlement.
With regard to 2. all your lawyer is doing is facilitating his/her ability to negotiate a settlement. Your lawyer will attempt to secure from the defendant the highest settlement offer he/she can get. All he is doing is setting a "floor." You agree that if x dollars is offered you give consent for him/her to take that settlement without first confirming it with you since you already agreed that you will take that amount.
Say for example you meet with your attorney and you are told 1,000,000 is a good settlement. If it was offered you would take it. Your lawyer then negotiates with the defendant and say makes a demand of $3,000,000 knowing that will never be offered. The defendant offers $200,000. Your lawyer will reject this amount since you have not given him authority to settle for anything less than $1M. If he gets the an offer of $2M he will take it since you authorized a settlement of $1M or more. Say this negotiation goes back and forth and the offer is $700,000.00 and the defendant will offer no more. Your lawyer cannot accept this amount--your lawyer would then go back to you and say the defendant will offer no more than $700,000.00 what do you want to do. He/she will give you advice as to whether this should be accepted or rejected and proceed to trial. Such an authorization is very common in these types of case.
As for 3, see my prior answer to your other question.
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I agree with Mr Weitz that these requests seem unusual. I don't think these are things you should bring up in open court. Your lawyer should advise you of the risks of going to trial but number 2 is a little concerning. But I do often document what a client has authorized me to do just to make sure we are communicating. You know what has transpired in the course if your case. Often things happen that make the likelihood of success greatly decreased. Do not assume that your lawyer is working against you. Consider what he is saying.