This specific part of the author's argument is economic. The argument seems to take roughly the following form:
Direct resource extraction will become more difficult as we exhaust the easily-accessible and are forced to extract deeper, further away, and more dangerously
More difficult --> more expensive
More expensive --> we will find and use cheaper ...
In Europe, the data show larger numbers than the claim.
According to European Topic Centre on Sustainable Consumption and Production:
Municipal waste represents approximately 14% of all waste generated.... [T]he largest fraction is paper and cardboard at 35% of the waste stream, followed by organic material at 25%.
The overall low percentage seems to ...
We have this from 2010:
(Facts as of print time for the article.) There's a plant in VA that produces annually about 4 billion of the little plastic bendy straws that come with juice boxes. It has more than 80% of the US market for bendy straws, and also exports to a ...
The answer is going to depend on the region.
In Australia, for example, The National Waste Report, claims that in 2006-07,
22 707 000 tonnes or 52% of Australia's waste was recycled
and that figure:
22% was from the municipal solid waste (MSW) stream.
(i.e. the recycle bins, rather than from industrial, construction, etc.)
That is, 11% of all ...
According to The Green Book (2007):
Each year, Americans throw away 138 billion straws and stirrers
So, if coffee stirrers are included, that is 378 million per day.
A 29 May 1924 Pulp and Paper Magazine of Canada article says:
No other country, it is stated, can compete with the United States in the manufacture of artificial straws, and no ...
This wikipedia article states that 40% of Swiss household waste is recycled (citing official administration statistics, I can't tell from your question whether you're likely to believe them).
And this site claims that 91% of aluminium beverage cans are recycled (in Switzerland)
You are not going to be able show such things, as there are many other reasons why forest area goes up or down apart from paper production.
If you want figures on changes in forestry in different countries, you could try FAO Statistics, such as this which suggests forested area has risen in Europe and North & Central America but fallen in other ...