What are the requirements from a Round function in Feistel network? Let's assume a balanced network with many (dozens?) rounds. And a simple (selecting key bits) key schedule. 
What does it take from the round F function to make a secure block cipher? obviously we need non linearity and some avalanche. But is this sufficient, how can this still fail?
Does a high round count compensate for any mild weaknesses in the F function?
What does it take from the round F function to make a secure block cipher? 


The ability of providing resistance to most common attacks like differential, linear, algebraic, slide and others mainly depend on the features of S-box (if used). * Simon does not use an S-box.
To design F-function, it depends on the structure of your Feistel
cipher Type I, II, III and others (which differ in term of number of
branches, number of F-function, internal structure of F-function).
The size of plaintext (64,128) could also influence the size of box
(4-bit or 8-bit), like piccolo vs Clefia. Therefore, the differential probability, linear probability and algebraic immunity differ, 8-bit S-box provide higher immunity features, therefore lower number of active S-boxes needed to provide minimum immunity, hence lower number of rounds.
The internal structure of F-functions such as SP vs SPS influences
diffusion (in term of number active-box). In addition, the use of
different MDS matrices in r-rounds (multiple round diffusion or
diffusion switching mechanism (DSM)) provide higher diffusion compared
to using 1 MDS matrix. CLEFIA employs DSM.
On the other hand, 8-bit s-boxes such as those in AES are expensive in term of
masking against side channel attacks. One way is to use smaller
s-boxes (easy to mask) to build larger s-boxes or using 4 bit S-box.


Does a high round count compensate for any mild weaknesses in the F function?


I don't think if you have self-similarity F-function, it will be
vulnerable to slide attack, increasing the number of rounds will not
providing compensate for weakness. To solve the self-similarity of F-rounds it is recommended to add constants. For how to chose round constants for simple key scheduling, I recommend to read 



  Proving Resistance against Invariant Attacks: How to Choose the Round Constants


There are others structures of internal F-function (not only SP, SPS) such as ARX (e.g. the Simon cipher) that have been design in a triangle of security, performance and size, and resistance to both mathematical and implementation attacks (such as timing, DPA).