In Mathematics, what is the standing of the journal Proc. AMS?
Scimago is a clearly false, possibly semi-fake ranking. It is very easy to check this: for instance when I clicked on your link I got the following amusing result:
Vital and health statistics. Series 2, Data evaluation and methods research is considered of higher ranking for mathematicians (!) than the Annals of Mathematics.
There is no need to elaborate more.
Subjectively my impression is that Proceedings of the AMS is a very good journal for short papers. Having a paper there says two things: 1) The paper is short, and 2) For a short paper, it's a good paper. No one is going to mistake a Proceedings publication for one in the Annals, but no one is going to question whether it's good work.
As for the rankings, very short papers are rarely as substantial or influential as longer papers, so it's not really fair to compare a journal that only publishes short papers to a typical journal. It would be more fair to compare it only to the short papers in another journal, but this data is difficult to find. Nonetheless I expect that Proceedings would rank reasonably highly once you compare papers of similar length.
I tried an alternative measure that seems reasonable, but in the end the numbers are weird so I don't think it's a great measurement. But since I have the data here it is.
One could rate journals per page instead of per article. Eigenfactor gives a nice way to do this, since the un-normalized Eigenfactor is calculated at the level of all papers published in the journal. Their "article influence" is measured by dividing Eigenfactor by number of articles (and normalizing), so we can equally well calculate "page influence" by dividing 2010 Eigenfactor by 2010 number of pages published. Normalizing by multiplying by 10^6, this gives: JAMS 13.4, Acta Math. 8.6, Proc. AMS 7.1, Annals 5.9, Trans. AMS 4.9, J. Alg. 4.9, Adv. Math. 4.5. By that measurement Proc. AMS would be a very highly ranked journal! I think that naively just measuring per page unfairly advantages short papers relative to long ones so one shouldn't treat these numbers as too meaningful, but the numbers you were looking at unfairly penalize short papers, and the truth is somewhere in between.
do you always submit somewhere else first, and it the paper is rejected then consider Proc. AMS ?
I've never submitted to the journal. In general, it is true that in many respects the best math journals are the generalist journals, but if a paper was rejected for not being interesting enough, I'd likely be substantially more inclined to resubmit it to a more specialist journal, and not to a generalist journal like the Proc. AMS. That said, if one is building up a standard career, it is important to build up a CV that has at least some papers which are in journals which are generalist journals, so people don't feel like your work is so specialized that anyone outside one's own area won't be able to interact with you. This is important for hiring decisions for post-docs, VAPs and tenure-track positions.
or maybe is publishing there an act of "giving back to the community by promoting the AMS" ?
I don't know of anyone who publishes with the AMS to deliberately give back to them. Unless one is already tenured, having that sort of desire to give back influence where one publishes would be odd. There are a bunch of other ways mathematicians give back to the AMS, such as by writing reviews for MathSciNet, which is run by the AMS.
or is it mostly a journal for anybody who needs a quick publication (as the workflow seems quite fast) ?
In general in math, one rarely needs quick publication since one can put things up on the arXiv and math is very fond of preprints. There is a difference between a preprint that's been submitted to a journal and an accepted paper, and if one is at a particularly bureaucratic school, such a thing might even be a relevant distinction for tenure if one is borderline, but that would itself be weird. While issues with long publication time can be a problem (I have one personal horror story involving it), it is very rare to target a journal based on their review/publication time. I'm also not sure what makes you conclude that the workflow is quick. What gives you that impression?