To what extent is hydroelectric power really renewable?
Electricity from waves, from hydro (both run-of-river and storage) and from wind, are all indirect forms of solar power. Electricity from tides is different, and we can deal with that in a separate question. Global tidal electricity generation is not yet at the scale of gigawatts, so it's tiny for now.

Winds come about from the sun heating different parts of the planet at different rates, due to insolation angles, varying cloud cover, varying surface reflectivity, and varying specific heat of surface materials. Temperature differentials create wind currents.

Waves come about from wind, so they're a twice-indirect form of solar power.

Sunlight on water speeds up evaporation, lifting the water vapour into clouds, giving them lots of gravitational potential. That rain then falls, sometimes onto high land, from where it can be gathered into storage reservoirs that are tapped for electricity, or where it flows into rivers that are then harnessed in run-of-river hydro.

How much power is there? Well, the insolation from the sun is, at the outer boundary of the Earth's atmosphere, at an intensity of about 1400 Watts per square metre. The Earth's albedo is roughly about 30% - i.e. on average about 400 Watts are reflected back into space, giving an average irradiation into the Earth of about 1000 Watts per square metre. Picture the Earth's surface as seen from the Sun: wherever the Earth is in its orbit on its own axis, and around the Sun, the Sun sees a disc that has the Earth's diameter, so the surface area exposed to the Sun is just $\pi$ times the square of Earth's radius, which is about 6 300 kilometres.

So the incoming solar radiation is $1000 \times 6,300,000^2 \times \pi \approx 125 \times 10^{15} \rm \ W$

Current human consumption of non-food energy is about 17 TW, i.e. $17 \times 10^{12} \rm \ W$, which is about four orders of magnitude smaller than insolation.

There is an interesting open academic discussion (see "modelling methodology smackdown" section) about how much energy we can physically extract from wind: some models say it's only a few times as much as all the energy we currently use; others say it's another order of magnitude more again. Either way, it's way more than we need, so while it's an interesting modelling discussion, the answer wouldn't change anything right now.

There are also open academic discussions about the impact of removing terawatts of wind energy, on regional climates. However, regional climates are already one of the more uncertain parts of long-term climate forecasts; add to that the high uncertainties around where the energy extraction would be, and what the specific effects of extraction are, and we end up with a lot of speculation and very little usable information. Not that this has prevented people from trying to model it, and reporting their results: but the uncertainties dwarf the results.

And harnessing renewables doesn't change anything significant in the Earth's total heat budget either, because almost all of that $1000\ \mathrm{W/m^2}$ from the sun already ends up as low-grade heat anyway. If harnessed for electricity, that electricity will then get used, and almost all of it will then end up as low-grade heat. The bit that will change when we get close to 100% renewables is that we will no longer be injecting about 15 TW of energy from fossil and nuclear into the biosphere (2 TW is renewables). But that's only about 0.01% of the insolation anyway.