*Short answer:* 1. Yes. 2. The meaning comes from its words.

**To get (a)round to** is a phrasal verb: The origin of the meaning is simply found in its words.

The [*Oxford English Dictionary*][1] defines both *to get around to* and *to get round to* as phrasal verbs, having the same meaning:

>*With **to***: to succeed in finding the time, energy, or inclination for (doing something); to come to the point of dealing with.

Without the *to*, the verbs diverge in meaning, as you can see at the end of this post.

>**to get around to**  

The earliest recorded usage in the OED is from 

>1887   M. E. Wilkins *Humble Romance* 35   There has been a good many things I haven't **got around to**.

followed by [P.G. Wodehouse][2] in [Laughing Gas:][3]

>1936   P. G. Wodehouse *Laughing Gas* xxvi. 269   ‘I want to know why you haven't tied him up.’.. ‘We was aiming to **get around to** it later.’

which is an interesting quote also because of **was aiming to**, which means

>To have (something) as an object, intention, or desired outcome; to be determined upon; to seek to achieve or obtain. 
>(a) With infinitive as object. Also simply: to intend, to mean (formerly chiefly *England regional* and *U.S.*, now *colloquial*).

The first usage is from 1400, so this goes way back. But it used to be a regional thing; it is now considered [colloquial][4].  

Before WWII Wodehouse lived in both the US and England; he often represented "the English" to American readers and vice versa, publishing books in both countries. Wodehouse had a [75-year career][5] as an author. Astonshing. Even in some of his later works he uses expressions found in his earlier works 70 years later, when the world had been a different place (before both world wars).

The first genuine usage for **get round to** is 1946.

>1946   K. Tennant *Lost Haven* (1947) xiv. 221   Everything in Lost Haven was put off until someone should have enough time to **‘get round to it’**.

Notice the quotation marks around the phrase; this indicates something about the attitude of the phrase by the narrator of the story. Perhap he means the phrase is new-ish or that its usage is ironic. (By the way, Wodehouse has a lot of uses of words with the ending -ish, showing that this suffix is not anything new, despite its recent resurgence in use.)

As Jasper has shown, we can use *can* before *get around to*, just like we can many verbs, with the sense of being less direct or more polite. To see this meaning of *can*, try it as an imperative (an order):

>You *can* sit over there. You *can* sit here. 

These are actually orders, but the *can* significantly softens them.

So, you know

>Why don't you upvote this answer and Jasper's, whenever you *can get around to* it? :)

----------


Without the *to*, the two phrasal verbs have **other** meanings:

**to get around** can mean   
>To go round; to circulate; *specifically* to go out a great deal; to visit many places. Chiefly *U.S.*
>
>1928   *Amer. Speech* 3 219   Get around, to..have many desirable dates. ‘Mary Jane sure did get around last semester.’

Note that today if we said "Mary Jane sure [gets around][6]," it would often (always?) implies that she *sleeps around* ([has sex with multiple partners][7], or is at least alleged to). So you can see how the meaning has changed from 1928 to today.

>1951   M. McLuhan *Mech. Bride* 60/2   The news got around fast.
>
>1959   *Times Lit. Suppl.* 20 Mar. 159/3   Still, Mr. Donnelly has got around... He makes his way to places like Tashkent, Samarkand and Alma Ata.


**to get round** can mean:
>To recover from illness, get well.

The idea, I imagine, is to get round a disease and make it back to health. But I can't recall this phrase actually used this way. Nowadays, we say **[come (a)round][8]** (apparently only an AmE usage) or, using a different verb **[pull through][9]**. 

>1857   T. Hughes *Tom Brown's School Days* ii. vi. 349   Did they tell you..that poor Thompson died last week? The other three boys are **getting quite round**, like you.

I wouldn't use this phrase to have this meaning, as it seems outdated, although the OED does not indicate it as such. But *come around* is used today, as is *pull through*.


  [1]: http://en.wikipedia.org/wiki/Oxford_English_Dictionary
  [2]: http://en.wikipedia.org/wiki/P._G._Wodehouse
  [3]: http://en.wikipedia.org/wiki/Laughing_Gas_%28novel%29
  [4]: http://www.macmillandictionary.com/dictionary/british/colloquial
  [5]: http://en.wikipedia.org/wiki/P._G._Wodehouse_bibliography#Chronological_list
  [6]: http://www.macmillandictionary.com/dictionary/british/get-around#get-around__5
  [7]: http://www.macmillandictionary.com/dictionary/british/sleep-around
  [8]: http://www.macmillandictionary.com/dictionary/american/come-around
  [9]: http://www.macmillandictionary.com/dictionary/british/pull-through