Do recent announcements about solving the DLP in $GF(2^{6120})$ apply to schemes proposed for cryptographic use?
The quoted recommendations do little to prevent fields that are subject to the recent developments. Take the $\mathbb{F}_{2^{6120}}$ example: it clearly passes the field size criterion, but also the subgroup rule, as the group order $2^{6120} - 1$ has one $1536$-bit prime factor.

Not all binary fields are affected equally, however. Both Göloğlu et al and Joux's approaches descend from the medium-prime function field sieve, and require the field $\mathbb{F}_{2^N}$ to be representable as (among others):


$\mathbb{F}_{q^n}$, where $q = 2^l$ and $n = 2^{l/k' \cdot d_1}$, $d_1, k'$ constant  (Göloğlu et al);
$\mathbb{F}_{q^n}$, where $q = 2^l$ and $n = 2^{l/k'} - 1$, $k'$ constant  (Göloğlu et al);
$\mathbb{F}_{q^{2n}}$, where $q = 2^{l}$ and $n \le q + \delta,\, \delta \ll q$ (Joux).


When the exponent of the binary field is smooth, we get many choices to try and represent the field in the forms above. Example: $1778 = 7 \cdot (2 \cdot 127)$; $6120 = 24 \cdot (2^{24/3} - 1)$; $6168 = 257\cdot (2\cdot 12)$; etc. When the exponent is prime, however, we have no such luck. 

Instead, as Joux suggested, one may embed the target field within a larger field that does: $\mathbb{F}_{q^{2N}}$, where $q = 2^{\lceil \log_2 N \rceil}$. While this solution may be asymptotically satisfactory, the blow up in size increases the bitlength where Joux's algorithm is faster than the regular function field sieve. It is currently unknown what this size may be, but it is likely that is is higher than current common key sizes (1024-4096 bits).

As for actual uses of binary fields, I do not know of any real-world usage of schemes based on the DLP over binary fields (beyond pairings). We have long been avoiding composite degree binary fields for elliptic curves due to their increased susceptibility to index-calculus; it seems that finite fields are now forced to do the same.