Random Puzzles 

Workings

Hint
There are quite a few anagrams, one of which is LEAST.

Answer
Least, slate, stale, steal, tales, teals.

Remember that the question asked for common words.

Workings

Hint
It will take a while to check 1 x 7, 2 x 7, 3 x 7. What do we know about the answer?

Answer
63,492 x 7 = 444,444

Reasoning
We can work backwards, as we know that the number multiplied by 7 has only 4's. We can try the following:
      4 ÷ 7 ≈ 0.57     x
     44 ÷ 7 ≈ 6.29     x
    444 ÷ 7 ≈ 63.43    x
  4,444 ÷ 7 ≈ 634.86   x
 44,444 ÷ 7 ≈ 6,349.14 x
444,444 ÷ 7 = 63,492

Workings

Hint
Another example, PEN is a complete word because:
   -en = ten
   p-n = pin
   pe- = pea

Answer
Three of the words are not complete: flew, rook, lamp.

Reasoning
The remaining 7 words can be changed as follows:
team:  -eam = beam
 t-am = tram
 te-m = term
 tea- = tear

cost:  -ost = host
 c-st = cast
 co-t = coat
 cos- = cosy

flew:  -lew = blew
 f-ew = ---- X
 fl-w = flow
 fle- = flex

ward:  -ard = card
 w-rd = word
 wa-d = wand
 war- = warm

fail:  -ail = nail
 f-il = foil
 fa-l = fall
 fai- = fair

rook:  -ook = book
 r-ok = ---- X
 ro-k = rock
 roo- = roof

soft:  -oft = loft
 s-ft = sift
 so-t = sort
 sof- = sofa

most:  -ost = cost
 m-st = mast
 mo-t = moat
 mos- = moss

lamp:  -amp = damp
 l-mp = lump
 la-p = ---- X
 lam- = lamb
Note that LARP is an uppercase acronym, and not in any of my word lists.

load:  -oad = road
 l-ad = lead
 lo-d = loud
 loa- = loaf

Workings

Hint
Think laterally.

Answer
In the ground!