You cannot receive a work permit as a TD.
You may be eligible for a 12-month OPT (Optional Practical Training) work authorization document upon completing your studies, or CPT during your studies. This would require you to be in F-1 status during your studies. Check first with your educational institution to see what type and duration of work authorization you might be eligible for based on the course of study you plan to take.
If you have a valid F-1 visa and valid I-20 at the time of re-entry into the US you should be able to return. It is a good practice to have additional supporting documents such as your financial documents in support of your I-20 and proof of nonimmigrant intent (i.e. your ties to your home country) in case you are asked to produce them.See question
All you need to do is to state the correct information there for the last five years. Start with the most recent employment and provide the employer's name and address, your position and dates you worked there, whether you worked in the US or in your home country. Even if the employment was brief (such as in the US), or much longer (such as in your home country). Don't worry about gaps in your employment, you can explain them if asked at the interview.See question
USCIS used to extend the premium processing courtesy to concurrently filed applications of the dependent family members. USCIS stated recently that this practice has been discontinued. Since the I-539 (H-4) filing fee check has not been cashed, there may be an issue as to why this application was not receipted. You haven't indicated how long ago the application was filed. Inquire with USCIS by telephone to see if you can obtain any information from them. The current H-4 status is valid through the I-94 expiration date in July 2019.See question
This should have no impact on her H-4 visa. She can try to apply again with evidence disproving her immigrant intent. Best to speak with an immigration lawyer about the circumstances of her refusal and the type of evidence she may present. However, it is not recommended that she shop for another consulate.See question
Provide the consulate with all and any evidence that you are settled in Sweden and have no plans to immigrate to the US in the foreseeable future (e.g. proof of employment, proof of ownership of real estate, bank accounts, memberships, family members who need your assistance, etc.). In addition, provide evidence of what your plans are during your short visit in the US and the reason(s) why you need to return to Sweden after your short trip.See question
The application will be returned to you and then you can re-file it properly with your check.See question
Can you get a sponsoring company to apply for an H-1B visa?See question
Talk to an immigration attorney so that s/he can help you to clarify the discrepancy to USCIS.See question
He needs to have a return ticket that is very important. He should have some document(s) demonstrating that he is your brother and the date of your graduation ceremony. These are the most important. Since he will be here for 10 days only, you can also give him a letter that he will stay with you and he should be able to demonstrate enough funds to support himself here (bank statement).See question