At first, ...

Initially Thor refuses ...

In most contexts that come to mind, I personally would put it to the end of the first phrase: Thor refused to do so at first, but he gives in...

Firstly is IMO dated and I would not recommend to use it at all... But I'm not sure, I'll ask Fowler when I get home and come back on this :-)

Das Komma ist auf jeden Fall nicht falsch und ich persönlich finde es optisch einfach schöner... auch was das Verständnis angeht ist es empfehlenswert.

My Fowler says:
"The preference for first over firstly in formal enumerations is one of the harmless pedantries in which those who like oddities because they are odd are free too indulge, provided that they abstain from censuring those who do not share the liking. It is true that [some references for the use of -ly]; but the boot is on the other leg now. It is the pendant that begins his list with first; no one does so by nature, it is an artificialism. (...)"

Well, live and let live. Anyway, what's important here: one should only use fist/-ly should when a second/-ly is to follow, thus only in lists/enumerations. Stick with dude and opine.

Zu unterscheiden ist zwischen:

At first [Zuerst und NICHT: Als erstes]... but then ...

und

Firstly,
Secondly, (mit oder ohne Komma)
Third(ly)
...
Das ist eine Aufzählung.

As Braunbärin explains, there are two different meanings. 'At first' and 'initially' mean that something was first one way, but then it changed. That's the meaning you need here, and the comma after either of them is optional.

'First' or 'firstly' introduces a numbered list of sentences, reasons, examples, etc. Those are both wrong here, but they (and second(ly), third(ly), etc.) do require a comma, it's not optional.

OT: I would check whether that's a modern edition of Fowler. In its crankiness it reads like the original Fowler himself, from almost 100 years ago; I doubt that Burchfield, the current editor, would still agree. In any case, surely most teachers and editors today prefer 'first' and regard 'firstly' as old-fashioned or pedantic.

Fowler: [...] those who like oddities because they are odd are free too indulge [...]

two to Toulouse

It's the 2nd edition, 1965. I have long wanted to get the 3rd edition and now I just read on Wikipedia that there is something even newer, 2009.

Pachulke, ja sorry, habs abgetippt, da ich ihn nur old-school gebunden besitze. :)

Fowler was not cranky but arch. His works make marvellous reading, provide a wonderful education in good writing, and, surprisingly, little of it is dated.

One is of course free to disagree with Fowler, but Burchfield (1996) does not seem to do so; he considers first and firstly equally acceptable.

It does seem, however, that the trend today is toward first; the authoritative Chicago Manual of Style (15th ed.) eschews firstly.

I agree that First, Thor refuses to do so, but he gives in when . . . . must be corrected to At first, Thor refuses. . . .

I would say that a comma after at first in this case is optional but probably a good idea.