I though that degrees of freedom for t test is the number of participants minus two parameters (means). However in the row for the equality of variance not assumed it could be for example 35, 598. Any explanation for that?
I think it's due to the way that df need to be adjusted when equality of variances is not assumed: There's a good explanation at: http://stats.stackexchange.com/questions/116511/explanation-for-non-integer-degrees-of-freedom-in-t-test-with-unequal-variances
Which program do you use? Is it possible that the program automatically uses an adjusted test (Welch I guess), when the assumption of equal variances is not met? This would explain the decimals for the dfs.
In SPSS you should see two rows, the first row with the "normal" t-test (including the Levene test) and the second row with the Welch test with adjusted df to correct for unequal variances. Does it look that way?
I attached an example how my output looks like (SPSS 23, german version).
It has nothing to do with a snipping tool. It is cropped accordingly. It does not work on my machine when I use "save graphic as", but "save target as". With the former command the download ist only 2kb, because it saves the thumbnail. With the latter command it saves the original file, which should be around 68kb.
I think according to APA you have to report 2 decimals, and this is reasonable. Maybe rounding would be ok if you additionally report the eta-value, i.e. the deviation from homogeneity, so that every reader could caclulate the exact df-value.