Why are there (still) coins? I find coins extremely inconvenient to carry and count. I feel I'm not the only one to find them annoying and inconvenient.

What are some economic reasons why all cash money isn't paper and coins still exist?
In 2013 researchers at the Federal Reserve Board published a paper {link} considering the costs and benefits of replacing \$1 bills with \$ coins.
Here are the takeaways from their executive summary: compared to notes, coins offer

Less payment system efficiency. Using \$1 coins instead of \$1 notes for transactions is inherently inefficient, requiring a replacement of one \$1 note with more than one \$1 coin to make up for the difference in the way coins and notes are used by the public.
Less cost-effective. Replacing \$1 notes with \$1 coins is also not cost-effective for the U.S. government and public more broadly, primarily because the higher cost to produce coins compared with notes is not offset by the longer life of the coin.
Higher costs under all projected scenarios. Circulating only \$1 coins costs more under every scenario we considered (in net present value terms) than would continuing to provide $1 notes to the public.
Increased costs to the private sector. Circulating only \$1 coins could result in increased costs to the private sector, perhaps in the hundreds of millions of dollars per year, and would more than offset any &quot;seigniorage revenue&quot; to the government reported in earlier GAO studies.

So the case for using coins is indeed weak. But reading the paper in more details offers some clues as to the historic reasons why coins are so widespread.
For example, low denominations tend to be used quite frequently in day to day transactions, meaning they wear out much faster. The UK, Canada, and Australia all replaced their equivalent of the \$1 bill with a coin in the 1980s. But this was at a time when electronic payment mechanisms (such as credit/debit cards) were not widely available, meaning many small transactions used the low denomination currency with high frequency. This lent an advantage to coins' high durability that is less important today.
Indeed, one of the historical advantages of coins has been their extra durability, which makes them cheaper to maintain in the long-run. But the FED report notes that the durability of notes has increased significantly. When the UK replaced its £1 note with a coin, the lifetime of a note was 10 months (many of the original coins are still in existence 30+ years later). But a 2012 \$1 bill has an expected lifetime of 70 months, tipping the economics of coins vs notes decisively in the latter's favour. Many countries are now adopting polymer (i.e., plastic) banknotes that have yet better durability.

In sum, notes appear to dominate coins both economically and practically (at least around the \$1 denomination value). Various countries have also eliminated lower denominations from their coinage altogether (see Wikipedia).
But there are costs involved in replacing coins with notes. The state must retool its currency production, provide public awareness campaigns, and administer the process of currency replacement; the private sector must modify vending machines, cash dispensers, and similar appliances, and incur other costs implied by the transition.
For example, The Guardian reports that it was cost up to £50m to replace just the nation's parking meters as the UK undergoes its switch to a new design of one pound coin. Another estimate is that it will cost £40-50 for each changing room locker that needs to be upgraded. A treasury estimate put the cost of introducing new 5p and 10p coins in the UK to the coin-operated industry at £80m.
These costs, along with general inertia and the fact the people increasingly avoid using cash altogether are probably the main reason why we still have coin currency.