You can be refused for any reason. A DWI is a felony offense (serious) in the US. You can absolutely be refused entry for a felony. I hope you bought insurance on your trip.
If it makes it any clearer, here is a copy of Condoleeza Rice's memo to all consulates regarding DUI/DWI convictions.
R 072132Z JUN 07
FM SECSTATE WASHDC
TO ALL DIPLOMATIC AND CONSULAR POSTS COLLECTIVE
RUEHTRO/AMEMBASSY TRIPOLI 3805
UNCLAS STATE 079496
E.O. 12958: N/A
TAGS: CVIS, CMGT
SUBJECT: GUIDANCE ON PROCESSING VISA APPLICANTS WITH DRUNK DRIVING HITS
1. Summary: This cable clarifies how consular officers should handle cases where an applicants' criminal record shows an arrest or conviction for drunk driving or other alcohol related offence. End summary.
2. Posts generally become aware of drunk driving arrests and convictions after receiving the results of fingerprints taken when an applicant has a CLASS hit. While a drunk driving conviction is not a statutory visa ineligibility, a conviction may indicate that further investigation is needed to determine whether the applicant may in fact be ineligible under Section 212(a)(1)(A)(iii). This applies to applicants who have a physical or mental disorder and demonstrate behavior associated with the disorder that may pose, or has posed, a threat to the property, safety, or welfare of the alien or others.
3. In the case of IV applicants, consular officers must refer the applicant back to the panel physician for additional evaluation. Physicians are evaluating for the presence of a mental disorder previously unnoticed before the physician became aware of the alcohol-related arrest. NIV applicants that have hits with evidence of an alcohol-related arrest or conviction must be referred to panel physicians for evaluation. This must be done even if the panel physician is physically located in another city.
4. After consulting with the Center for Disease Control and Prevention, we have determined that consular officers must refer applicants to panel physicians in two circumstances: (1) an applicant has a single drunk driving arrest or conviction within the last three calendar years or two or more drunk driving arrests or (2) convictions in any time period. Consular officers must also refer applicants to panel physicians if there is any other evidence to suggest an alcohol problem. Consular officers must adhere strictly to these guidelines in determining when a panel physician referral is appropriate.
5. For a finding of eligibility under Section 212(a)(1)(A)(iii), there must be two criteria established by the panel physician: (1) diagnosis of mental disorder (alcohol abuse) and (2) current harmful behavior associated with the mental disorder or a history of harmful behavior associated with the mental disorder that is judged likely to recur in the future. Consular officers should be aware that neither alcohol abuse or (DWI) drunk driving are sufficient grounds for an ineligibility finding under Section 212(a)(1)(A)(iii), a panel physician evaluation is required.
6. Section 9 FAM 40.11 N8.3 will be updated as follows:
While alcoholism constitutes a medical condition, INA 212(a)(1)(A)(iii) does not refer explicitly to alcoholics or alcoholism. Evaluation for alcohol abuse or dependence is included in the evaluation for mental and physical disorders with associated harmful behavior. An alcoholic is not ineligible to receive a visa unless there is harmful behavior associated with the disorder that has posed, or is likely to pose, a threat to the property, safety, or welfare of the alien or others. To ensure proper evaluation, you must refer applicants to panel physicians when they have a single drunk driving arrest or conviction within the last three calendar years or two or more drunk driving arrests or convictions in any time period. You also must refer cases to a panel physician if there is any other evidence to suggest an alcohol problem.
There is no law about who can or can't pay for a defendant's attorney in a criminal case.
This should be legal.