your employer should have all the documents for the company you are working for, what embassy is asking is the legitimacy of your employer company.....
NO, but logically something to support your stay in Thailand makes sense, if you say i get 200 or 400$ a month then they will think how will you support yourself....think how you would react if someone comes to you and ask you to stay in your house but not enough monthly income?
remote work means you are working so logically you are receiving money and you should have either payslip or payments from contracts coming to you, if you apply for soft power then they will not ask for payslips
normally they do not but if they do tell them you work remotely so your plans can change from 1-2 months, they should always give you 180days stamp for DTV.....so if they do give you less then ask them politely to correct ...dont leave the counter before you confirm the date on stamp on your passport is correct
NO airport and all the airports, if you have nothing to hide and able to answer there question honestly then trust me you are fine.....unless some IO is just in a bad mood and still will just give you had time but will let you in.....