Hi Louis, great question! Here are my thoughts.

How many rats are there in cities

I made some back-of-the-envelope calculations of this a year ago, the goal was to compare the cost of contraceptive treatment per rodent with the cost of contraceptive treatment per pigeon.

Not much has been published on this question, but there seem to be about 220 rats per $k{m}^2$ in Baltimore, if one extrapolates from the numbers presented in this paper.

Here are some other reports to cross-check against:
According to this report from a US agency, "the total rat population of New York City is estimated to be not more than 250, 000 or one rat for 36 persons". A similar claim is made here. Going by the ratio of rats to humans, both of these seem to be at least roughly in accordance with the Baltimore data, which is 47 000 rats and 619 000 humans. According to this source, where they try to estimate the density of rats in urban areas of the UK, "each rat has a rather spacious 5,000 square metres to roam around in." That means that there are 1 rat for every 0.005 $k{m}^2$, which is equivalent to 200 rats per $k{m}^2$. Also very close to the numbers for Baltimore.

could this be a potential area to improve animal welfare?

I think fertility control for rodents is promising, here are some somewhat related thoughts I have on contraceptives for pigeons. Although the only commercially available product for rodents that I'm aware of (ContraPest) is quite expensive as far as I can tell; see my back-of-the-envelope calculation.

I have been researching sterilizing rodents instead of killing them to control their populations, and it's much more popular already than I had realized. ContraPest is a bait that sterilizes rats with a few doses. It reduces sperm viability in males and induces aging of ovarian follicles in females, sort of like early menopause. There's a bit of a lag before the population reduces, but it has the benefits of humaneness, not disturbing the rats' territories (because older rats stick around, preventing movement between territories which can spread disease), and providing a better longterm maintenance solution. It's already widely used, and Senestech, the company that makes it, has had big contracts with cities like NYC and Wasington DC. 

I was very surprised to find out how widespread the use of sterilants already was considering I had not heard of them for rodent pest control until last year!

I think this is a good cause not only to reduce harm to household pests, but because having to participate in cruelty toward animals can lead to cognitive dissonance or defensiveness or the status quo treatment of animals. Finding out about sterilants got me out of binary way of thinking towards rat infestation (it's them or me) and that's the kind of creative problem-solving we need if we're ever going to make real improvements in wild animal welfare. 

Yes, I think mice are a potential area of animal welfare improvement. Charity Entrepreneurship wrote a report on an intervention for campaigning to ban the sale and use of glue traps, but they concluded it's not as cost-effective as donating to or working for existing top animal  welfare charities:

"Ultimately, we do not recommend banning the sale and use of glue traps as a way to improve welfare. Based on our research, this intervention does not look promising when considering counterfactuals due to the small amount of suffering prevented by averting a glue trap death. This is mostly because the alternative methods of pest control that would be most commonly used after a ban on glue traps appear to be rodenticides and snap traps, and death from rodenticides seems worse than from a glue trap. 

Although when compared to a glue trap death, a snap trap death looks better than a rodenticide death looks worse, the average amount of suffering averted is dragged down by this likelihood of replacement. Based on this and on our analysis of the cǒfounder counterfactuals, we do not recommend this charity get started this year. However, we think that it might be worth trying to convince an existing organization to do this intervention themselves.

Also, their report also has a section on why snap traps are better than live traps or glue traps:

"Animal protection organizations ̈such as the RSPCA advocate for the use of snap traps over live traps and glue traps, and many pest control organizations also recommend using snap traps over glue traps."

There has recently been some really exciting progress on mouse and rat fertility control. There are several products already on the market that claim to reduce fertility but they could certainly use more development and third party rigorous testing to show effectiveness. I think that investing in developing, testing, and promoting fertility control methods is likely to have a high payoff. Doing so is also likely to be profitable to private companies and in the interest of various governments, so this is a good opportunity to leverage resources of larger non-animal-welfare focused institutions.