Hello and welcome to our forum. Thank you also for your kind words.
Everything works as it should. When the select field is called and number of select option is large, the time to load the entire list (could be thousands) and even the process of selecting the record is rather painful and slow. EPESI automatically shows the select choice and it behaves normally when the list is short. I believe we set this value to 60, maybe 50. Once the list is large the input field automatically becomes what we call auto-select. It is Ajax driven select and you can type portions of the searched value, for example looking for "Joe Doe" noth "Bill Doe" you can type "Doe J".
The same is true for our multiselect field, like in the field Assigned Employees used in many modules. When the list is over a certain safe limit it becomes auto-multiselect field and you also type fractions of searched value to return possible matches.
If this is not happening on your installation then probably you have some javascript error which prevents the code from executing. If you are using Chrome open development tools and debug it here. You should see javascript errors in the console. Paste it here and we'll see what happened.
"And when typing nothing happens." - go to our demo site and see if we are talking about the same thing: http://epe.si/live-demo
" If I delete the newly created company the dropdown list works fine again, until new company entry is saved again." - yes, because with newly created company you just exceeded the safe limit for select field. You delete it and you are back within the safe limit and auto-select becomes back a normal select field.
Please note, that in many modules the auto-multiselect field allows combination of multiple record sets, like CRM Companies (marked as [C] for Comapany) OR CRM Contacts (marked as [P] for Person).