Incremental strongly connected components
To the best of my knowledge, the best algorithm for decremental strongly connected components is presented in [1] with $O(m \sqrt{n} \log n)$ total expected update time. 


  [1] Decremental Single-Source Reachability and Strongly Connected
  Components in Õ(m√n) Total Update Time - Shiri Chechik, Thomas Dueholm  Hansen, Giuseppe F. Italiano, Jakub Łącki, Nikos Parotsidis -  https://ieeexplore.ieee.org/document/7782945 - FOCS 2016


Update: The new best Algorithm for decremental SCCs is $O(m (\log n)^4)$ presented in [5]


  [5] Decremental Strongly-Connected Components and Single-Source Reachability in Near-Linear Time
   - Aaron Bernstein, Maximilian Probst, Christian Wulff-Nilsen -https://arxiv.org/abs/1901.03615 - accepted at STOC 2019


The best algorithm for incremental strongly connected components is presented in [2] with $O(m^{1/2})$ update time per edge.


  [2] Incremental Cycle Detection, Topological Ordering, and Strong
  Component Maintenance - Bernhard Haeupler, Telikepalli Kavitha, Rogers
  Mathew, Siddhartha Sen, Robert Endre Tarjan -https://arxiv.org/abs/1105.2397 - 2012 ACM Trans. Algorithms


It is an open question if those two problems can be solved faster.

For fully dynamic strongly connected components conditional lower bounds are known. Worst-case lower bounds are also known for decremental/incremental strongly connected component maintenance. For details look in [3] and [4].


  [3] Unifying and Strengthening Hardness for Dynamic Problems via the
  Online Matrix-Vector Multiplication Conjecture - Monika Henzinger,
  Sebastian Krinninger, Danupon Nanongkai, Thatchaphol Saranurak -  https://arxiv.org/pdf/1511.06773.pdf - STOC 2015
  
  [4] Popular conjectures imply strong lower bounds for dynamic problems
  - Amir Abboud, Virginia Vassilevska Williams - https://arxiv.org/pdf/1402.0054.pdf - FOCS 2014


All of these results hold for general graphs. Better results are known for planar graphs.

The result you mention about DAGs is well known:


  Amortized efficiency of a path retrieval data structure - G. F. Italiano - https://www.sciencedirect.com/science/article/pii/0304397586900988 - TCS 1986