This is from Buell, *Binary Quadratic Forms*. From page 84, the class number for a negative discriminant $\Delta$ is about $$\frac{\sqrt{|\Delta|}}{\pi},$$ which comes from an $L$-function calculation on page 83. 

Let's see, on page 101, he points out that for negative field discriminants, class group and narrow class group are identical. Then on page 103, the group of classes of binary quadratic forms is isomorphic to the narrow class group. So that works out. 

I don't know about surjectivity of class numbers.  I imagine so. See [OEIS][1] 

I wrote a little program up to 1000, here it is up to 111. The first number that achieves a given class number tends to be squarefree, an exception being h=104. 

        1       3 = 3
        2      15 = 3 * 5
        3      23 = 23
        4      39 = 3 * 13
        5      47 = 47
        6      87 = 3 * 29
        7      71 = 71
        8      95 = 5 * 19
        9     199 = 199
       10     119 = 7 * 17
       11     167 = 167
       12     231 = 3 * 7 * 11
       13     191 = 191
       14     215 = 5 * 43
       15     239 = 239
       16     399 = 3 * 7 * 19
       17     383 = 383
       18     335 = 5 * 67
       19     311 = 311
       20     455 = 5 * 7 * 13
       21     431 = 431
       22     591 = 3 * 197
       23     647 = 647
       24     695 = 5 * 139
       25     479 = 479
       26     551 = 19 * 29
       27     983 = 983
       28     831 = 3 * 277
       29     887 = 887
       30     671 = 11 * 61
       31     719 = 719
       32     791 = 7 * 113
       33     839 = 839
       34    1079 = 13 * 83
       35    1031 = 1031
       36     959 = 7 * 137
       37    1487 = 1487
       38    1199 = 11 * 109
       39    1439 = 1439
       40    1271 = 31 * 41
       41    1151 = 1151
       42    1959 = 3 * 653
       43    1847 = 1847
       44    1391 = 13 * 107
       45    1319 = 1319
       46    2615 = 5 * 523
       47    3023 = 3023
       48    1751 = 17 * 103
       49    1511 = 1511
       50    1799 = 7 * 257
       51    1559 = 1559
       52    1679 = 23 * 73
       53    2711 = 2711
       54    2759 = 31 * 89
       55    4463 = 4463
       56    1991 = 11 * 181
       57    2591 = 2591
       58    2231 = 23 * 97
       59    2399 = 2399
       60    2159 = 17 * 127
       61    3863 = 3863
       62    2471 = 7 * 353
       63    2351 = 2351
       64    2519 = 11 * 229
       65    3527 = 3527
       66    3431 = 47 * 73
       67    3719 = 3719
       68    2831 = 19 * 149
       69    3119 = 3119
       70    3239 = 41 * 79
       71    5471 = 5471
       72    3311 = 7 * 11 * 43
       73    2999 = 2999
       74    4151 = 7 * 593
       75    4703 = 4703
       76    3071 = 37 * 83
       77    6263 = 6263
       78    5111 = 19 * 269
       79    4391 = 4391
       80    5183 = 71 * 73
       81    3671 = 3671
       82    3839 = 11 * 349
       83    3911 = 3911
       84    4031 = 29 * 139
       85    4079 = 4079
       86    6767 = 67 * 101
       87    5279 = 5279
       88    4199 = 13 * 17 * 19
       89    6311 = 6311
       90    5951 = 11 * 541
       91    4679 = 4679
       92    4991 = 7 * 23 * 31
       93    5351 = 5351
       94    7367 = 53 * 139
       95    6959 = 6959
       96    6071 = 13 * 467
       97    5519 = 5519
       98    6191 = 41 * 151
       99    5591 = 5591
      100    7991 = 61 * 131
      101    5879 = 5879
      102    9383 = 11 * 853
      103   13799 = 13799
      104    9359 = 7^2 * 191
      105    6719 = 6719
      106    7631 = 13 * 587
      107    8231 = 8231
      108    5759 = 13 * 443
      109    5711 = 5711
      110    7751 = 23 * 337
      111   15359 = 15359 


  [1]: http://oeis.org/A060649