Ok....that added bit of info from you has me thinking I may have a solution for you. But more about that later....
Again...
Valve adjustment...
....has that EVER been done on this 20 year old outboard?
If you don't know then it at LEAST needs to be checked.
And, yes, you're right, you need to learn how to do it because it's VERY important. Not hard, but important.
If the valves get too "tight" due to wear, the engine could be badly damaged.
You'll probably need a service manual to do that. And, with all the slipshod, fly-by-night, flat out rip-off, so called "repair shops" out there STEALING people's money and RUINING their outboards....
...I urge you to get that manual and do the work yourself.
You can buy the AUTHENTIC shop manual here:
publications.powerequipment.honda.com
The manual is expensive but worth every penny if you care about maintaining that sweet 15 and staying out of the clutches of the flim-flam outboard repair pirates.
The "discount alternative" manuals like Seloc, Haynes, Clymer and others aren't worth the powder it takes to blow them to Washington D.C. so don't be tempted.
Besides information you will need a standard set of feeler gauges and a very tiny adjustable wrench.
Read what I wrote about checking the pump for leakage in post #8 of this thread. You do that while the rocker cover is off and AFTER checking/adjusting your valves.
Now, back to your hot start problem.
When you say opening the throttle helps that indicates that it may be in a "partial flood" condition. That says to me that there's a problem with the "choke".
I put quotation marks around choke because this carburetor doesn't have a conventional butterfly type choke. It has a cable actuated "enrichment valve" instead. The "valve" is nothing more than a pellet bonded to the end of the choke cable.
I don't like this system because, when the cable housing gets worn the valve won't work properly. You cannot adjust it either. The only solution is to replace the entire cable assembly.
And, to top it off, the parts book is misleading. It shows the cable....item 17 in the link below and the "valve set, starter"....item 11... as being just the housing.
But....if you buy item 11, you get the housing AND the cable as an ASSEMBLY.
Believe me! ONLY BUY ITEM 11!
DO NOT buy item 17!
Buy OEM Parts for Honda Outboard 15HP Carburetor (Manual) Diagram
www.boats.net
I can tell you from experience that the cable (item 17) can NOT be easily mated to your existing "housing". Anyway, your housing is likely causing this problem as it is so trying to reuse it with a new cable would be futile. Just purchase item 11 and carefully install it.
I say carefully because the threads are plastic and are easily stripped. So don't overtighten.
You asked if starting the engine with 1/3 throttle would hurt and I will say probably not as long as you immediately close the throttle on start up.
But....running the engine rich all the time with the choke partially open will eventually cause issues with carbon build up and plug fouling. That along with excessive fuel consumption and the hassle of fiddling with the throttle every time you want to go makes replacing item 11 worth the trouble and the $26.48 + tax+shipping.
BF15D....one of the BEST twin cylinder outboards ever produced! You own a GEM!
Good luck.