Here are 3 snips to show the Adding search engines part, your 1st query.
Now, to answer your 2nd question...
Instead of having Startup as New tab, select Use Specific pageS & Add the site or sites desired. The example ⬇ is telling it to be the Bing Search Engine page.
NB: A wee 'heads-up... Take "Startup" literally. The 1st time the browser is opened it will go where "Startup" page is set to be. Every subsequent opening of it from Start menu or its Rt Clk Jumplist, New Window or Microsoft Edge Dev/Canary. The initial opening is all well & good and the engineers are working resolve the behavior with the subsequent openings... except for InPrivate, should all go to the assigned page(s). Startup should equate to any new opening of the browser. With Edge, if it's opened via its Taskbar icon & then, you go open it @ the Start menu, BOTH open to the set page not, the 2nd opening to default New tab. Same with the 2 clickables on its Jump-list, STILL, goes as User configured. To repeat, it is now a known issue which, the Team is working on resolving... FIX ETA unknown, right now. But, this DOES give what you sought; albeit, 'limited' (to just the FIRST new opening, only) at the moment
I already have set all the above mentioned setting as you mentioned. What I was talking about was when opening a new Tab, by means of the plus (+) key. It still opens the default Microsoft Home Page, with the Bing search engine.
It would be nice if it would default to a user selected Home page.
"It would be nice if it would default to a user selected Home page."
This is a known issue, that it does not respect the assigned page, after the 1st time being opened. It has been reported & the Team is working on a resolution. The can set the Startup page to say, MSN.ca and it WILL go there, BUT, only on the initial opening. If you open another instance, without closing the, already, open instance 1st), every next instance opened will > New tab. They are, still, trying to fix it.