First off the 30 days you are here now that is just ending is not a visa. It is an Exemption from not having bought a visa prior to coming to Thailand. The 30 Day Exemption stay that you are on now can be Extended another 30 days at a Thai Immigration Office for 1900 Baht.

If you go to Vietnam for a few days without doing what I just wrote above... When you come back you will be stamped in with another 30 Day Exemption Stay ... (not a visa).

Note: There is no 30 day Visa...

With onward tickets days after you land - I can't see any problems

Work Permit Photo - This is similar to any face / shoulders passport / visa app photo only a special size - 3 x 4 cm - blue background --- and the photographer will use photoshop to overlay a suit jacket and tie...

I agree Robert a visit would solve the problem. I just was not going to assume that the Immigration Office was nearby - some Farangs in Thailand have to travel a long way to the Immigration office. I wonder if the Immigration Office staff would clear this up with a phone call? Especially with a Thai speaking partner.

But I see that my habits may not apply to the OP's situation. However, a 3 week window of compliance does allow for an approximation of the next report date most times... IMO

I have never encountered that before but my dates while not 100% synced they are very close - more than close enough to fall within a two week before and one week after window if I had to guess because of a lost report slip. But I have always done my Extensions within 2 days or so of Extension expiry which coincided very closely with the 90 day report schedule over time. Also I had always done my 90 day report on the exact day scheduled.

If you can estimate the date of the last 90 day report down to a 2 week spread... Then go to immigration and check... Remember you can report 2 weeks before the date and one week after. Another better way - you have a stamp in your passport - the date you were given an Extension of Stay... Check that and count 90 day intervals...

Your friend heard wrong. As stated above in the first comment - there are no 15 day visas. A British Passport holder with a history of B Visas means he likely does not have any history of holding a string of tourist visas. Which is in his favor. Your friend should have cash on hand of 20,000 Baht to show to I/Os upon Reentry to demonstrate the ability to support himself and copy of a rental agreement or hotel reservation in Thailand. It is possible they may want proof of a flight / bus out of Thailand...

One of the problems that I have picked up on - is that the officers at the issuing Consulate do not fully explain the nature of M - Multi-Entry to the buyer of the METV. I have seen numerous posts where the holder of the METV thought that they had bought a 6 months visa TO STAY inside Thailand for 6 months... They get here to Thailand and do not have a clue that they will have to Exit and Reenter to get another 60 day stay each time and that there has to be successive border bounces to get more stays..

Local immigration offices sell Reentry Permits