I guess I'm one of Tracy's '2 experts'... Am working on a project to produce toolkits for anyone interested in identifying & investigating the archaeology of, and within, woods.
Hammer ponds - generally quite large, and often still hold water, but smaller ponds, 'penstocks', could be created to feed the main hammer pond by embanking the sides of streams up to a mile away. Sometimes all that can survive of these is the dam, or bay, that held the water back, now breached so the pond is dry. Doesn't sound like the depressions you have though - these seem more like mine pits that have filled with water over time. Are the hollows round or irregular? Do they have spoil mounds around them? Is there any sign of an access track to them which might point to shifting of extracted material? Don't know what the underlying geology of your wood is, so difficult to guess what might have been extracted. Could be marl pits if on clay? Other materials regularly extracted which can leave hollows might be stone, (small scale just for local use), or iron ore.
Researching your wood - where to start?! A huge subject, but a few pointers:
The Historic Environment Record (HER) - a database of all known finds and sites in each county, maintained by the archaeology team in each county council. Can provide specific info about your wood (if anything is recorded there), but just as importantly, give you an overview of how the landscape surrounding your wood was used over time. Also have a collection of aerial photographs back to 1940's which can show development of land use.
Record Offices - for parish & church records, & importantly the estate archives of local landowners with quite detailed past woodland management records. Account transactions for estates have fascinating info in them about woodland management - the larger the landowner, the more chance you have of tracing the history of a wood. Charters, deeds, manorial records may be in the RO, or may be held in collections elsewhere.
www.a2a.org.uk (Access to Archives) - can indicate archive collections you might look in - not all RO's have put their catalogues on this though.
www.nationalarchives.gov.uk/catalogue - lists material held at National Archives at Kew.
Local Studies or reference sections in libraries - trade directories (useful for small industry), previously published histories of the area, maps, census records (you may come across a family mentioned as working wood in some way in your area).
Local archaeology or history groups - may know something relevant to your wood. www.local-history.co.uk/groups
www.cartography.org.uk - gives info about collections of maps
www.viewfinder.english-heritage.org.uk - old photos of local history
www.pastscape.org.uk - national record of archaeological sites but not as useful as HER
Phew - lots more I could say but run out of time! Hope this helps..