I have a small planet with a Radius of 3,900 kilometers and a density of 8.9 g/cc. It has an extremely Earth-like climate and has been extremely volcanically and geologically active in the "recent" past, creating numerous fissures, lava tubes, and the like. This planet will only have a few freshwater lakes about the size of the Mediterranean sea, no large salty oceans. Is it feasible through a network of underground aquifers and rivers to have a planet largely (at least 85%) forested?
Rainforests generate their own rain
Forests, if extensive and active enough, can generate their own rain. Plants transpire, which is the movement of liquid water from the roots to the leaves of plants, and which results in evaporation from the leaves.
If you have enough plants in the same place, their transpiration can cause humidity to rise enough to cause cloud formation. If clouds form during the day, and the temperature drops (as, in the evening), then rain will fall. This will be the same water molecules that were transpired by the trees earlier in the day.
Depending on atmospheric circulation, it is certainly possible for a planet to be so heavily forested that all the rain that falls was previously transpired by those same trees. In essence, it is possible for a planet to be completely forested, and for its forested condition to cause it to rain every single day.
If the forests generate their own rain, then, given time, an entire planet could be covered with such forests.
The Earth-Like climate is incompatible with the lack of oceans. Earth-like climates rely upon ocean currents to move massive amounts of heat from the tropical zones, warming chilly places like Europe and western North America to temperate. Without ocean heat movement, you will have more clear, discrete boundaries between the tropical, temperate, and arctic zones. The arctic zone will be bigger than Earth's, and the temperate zone smaller. There will be no seasonal monsoons or cyclones/hurricanes/typhoons.
Evaporation, mostly from oceans, provide 86% of the water in the Earth's air. Even quadrupling the amount of transpiration cannot make up for the lost evaporation. That means less rain: Winter snowpacks will be thinner, rivers will be smaller, erosion and deposition will be slower, glaciation will be much less pronounced on landforms. That means fewer and less-widespread porous strata like limestone and sandstone, and relatively more non-porous strata like granite. In turn, that means smaller and fewer aquifers, and fewer springs.
Less rain, smaller rivers, fewer springs, and smaller aquifers means that the planetary surface will be relatively dry. Less polar glaciation means more hills and valleys (instead of flat plains) in the temperate zone. You will see more bigger trees where the water table is high in the valleys, but transitioning to mere scrub just a few meters up the slope away from that water table. With forests chopped up by hill/valley terrain, rainforests seem likely to be scattered and small.
Instead of competing for light, plant life will compete for water - trees in the wide marginal areas will be broad and short and spaced far apart, scrubby above ground while pushing deep roots in search of water, rather like alpine or sub-arctic trees.
With reduced seasonal rains (or snowmelt), steppe grasses and forest undergrowth will be reduced.
Given the information stated, this is not an Earth-like planet. Under many definitions it is a super-Earth. Diameter is larger than any terrestrial planet in the solar system and surface gravity is 2G. The density is impossibly high - it is more dense than iron, and iron is the final abundant product of stellar fusion.
Secondly, it is unavoidable for the final outflow bodies to be salty unless there is (somehow) no salt on this planet - which means no metal salts. Chemistry that weird immediately stops it being an Earth-like planet.
Finally, without some bizarre characteristics of the primary and orbit, there will not be a uniform climate and associated ecology across an entire planet. If you want all of the land mass to have the same climate then it all needs to be at roughly the same latitude under most circumstances. Star Wars has popularised the idea of a "desert planet", an "ice planet" and a "forest moon", but actual worlds will inevitably have temperature variations between different areas based on the sunlight received.