1. How compatible is it with the 3.5e stuff?
d20Modern, rules-wise, is maybe ... 3.25e. It has some changes from 3.0 that went into 3.5, but has some rules from 3.0 that were changed in 3.5.
Some rules in d20Modern are totally different from 3.X. For instance, d20Modern Nonlethal damage is a "check and discard" system. You compare the damage done to the Massive Damage Threshold of the character, if it doesn't meet or exceed it, the attack has no effect. This is pretty controversial and, IMHO, a "cinematic" rule but one many people discard in favor of accruing subdual like D&D.
Note also that D&D has a "Massive Damage Threshold" ... of 50. In d20Modern the MDT is set to your Constitution score.
2. Can I use monsters from the Monster Manual in my adventures?
Yup. In fact, if you check the back of your d20M book, about half of the things there are things ripped pretty much wholesale from the MM. There are some minor changes, but nothing you'll notice in day-to-day play. Urban Arcana pretty much meshes D&D with d20M.
3. Could I use, say, the races books or environment books?
I don't see why not. The rules are pretty much compatible, and the real differences are in what skills are available and running combat. You may find, for instance, that a race gives a bonus to a skill that doesn't exist in d20Modern, or that one of the environment books talks about using a skill that doesn't exist in d20Modern. Some have similar matches in d20M ... like the Heal skill in D&D and the Treat Injury skill of D20M.
4. Can my players run a DND character in a d20 Modern campaign?
I wouldn't suggest mixing d20Modern and D&D characters together. They don't mesh well. In many ways, d20M characters are less powerful than d20M characters ... in some ways, more powerful. It'd cause odd power gaps, especially in combat. D&D characters would own the day, especially with easy low-level magic and armor/weapon profs, along with many D&D classes having full BAB ... a D&D character would hit harder and have more options at first level, while a d20M character will have a better AC (Defense in d20M, and aided by Class Bonus to Defense) and many more skills and skill points. D20M characters are better at NPC interactions and problem-solving ... D&D at killing things and taking their stuff.
Note also that D20Modern has something called "Advanced Classes". They're between a Base Class and a Prestige Class. The d20M core book has "Mage" and "Acolyte" AdCs in the back that present the d20Modern "take" on the Wizard and Cleric, respectively. The good news is, many of the classic D&D archetypes are redone as AdCs. The bad news is ... they're spread across several books. IIRC, between D20 Modern Core Book, Urban Arcana, and d20 Past you'll get AdCs that replicate Wizard, Sorceror, Cleric, Ranger, Druid ... Barbarian ... Mmmmm. Can't remember, really.
If you're looking to do a "D&D Stuff In The Modern Day" campaign, look at Urban Arcana. That's ... it. Literally. They call it "from Shadow", but the setting is all about stuff from Greyhawk getting picked up and plopped into our world ... dragons with megacorporations, Drow running the media ... goblins mugging people in the subways and meeting mysterious benefactors at the local "Prancing Pony" in a dark corner booth.
Personally, its pretty hokey ... but a friend of mine ran an UA game and it was alot of fun.
But I'm not admitting that in public.
--fje