I'll address 3 and 4. Put new mulch on top of old. No need to remove. Mould is natural and harmless. Ignore it.

As to timing and material, that's a question of your climate, personal preference, convenience and cost. I mulch any time with anything I have to hand. I've never bought mulch and I've never had anyone do it for me.

Obviously, it is easier to mulch when plants are emerging from dormancy, if that happens where you are, but I don’t think it matters too much either way. Just don’t exceed a depth of about three inches, four is too much.

In Virginia with 40+ inches of annual rainfall, I need to mulch twice a year because the mulch breaks down. There's no question about putting new mulch over old: two inches of mulch applied in March is gone by October. If your mulch persists all season & even gets moldy, your landscaper is applying too much.

1) I mulch twice a year - early spring and in mid to late fall

2) as long as it is an organic product (bark mulch, compost, leaf mold, etc.), there is never a need to remove the old - just apply the new mulch on top

3) the "white mold" is most likely fungal mycelium. It is a perfectly natural sign that the mulch is being broken down or decomposed by fungal organisms. It is harmless. Sometimes you will get even more unusual fungal growth from a mulch product........it is still harmless :-)

4) types of mulches - or brand names - will vary from region to region. Use whatever is common or popular to your area and is of good value. Avoid dyed mulches (hideous!), those made from recycled rubber products or non-renewable materials (cypress mulch). If you need a quantity of mulch, bulk will always be far less expensive than bagged.

FWIW, less than 2 inches of mulch is not worth the effort. 3-4 inches is ideal in that it is the right thickness to retain soil moisture, protect against temperature swings and offers the best weed suppression.