For (3) consider the function $$f:\mathbb{R}^2\to\mathbb{R}:\langle x,y\rangle\mapsto \begin{cases}\frac{2xy}{x^2+y^2},&\text{if }\langle x,y\rangle\ne\langle 0,0\rangle\\\\
0,&\text{if }x=y=0\;;
\end{cases}$$ $f$ is easily seen to be discontinuous at the origin, but all of its sections are continuous.
Added: For background and references on separate versus joint continuity see Z. Piotrowski, Separate versus joint continuity - an update, Proc. 29th Spring Conference Union Bulg. Math. Lovetch (2000), 93-106, available here as Nr. 51.
T.O. Banakh, O.V. Maslyuchenko, & V.V. Mykhaylyuk show in Discontinuous separately continuous functions and near coherence of $P$-filters that for every non-discrete Tikhonov space $X$ there are a Tikhonov space $Y$ with a unique non-isolated point and a bounded separately continuous $f:X\times Y\to\mathbb{R}$ that is not jointly continuous and use this result to show
Theorem: For any non-discrete Tikhonov space $X$ there are a $\sigma$-bounded Abelian topological group $G$ and separately continuous $h:X\times G\to \mathbb{R}$ that is not jointly continuous.
The paper is available here as Nr. 125.
For (1), Maxim R. Burke, Borel measurability of separately continuous functions, Topology and its Applications 129 (2003), 29–65, mentions that Walter Rudin, in Lebesgue’s first theorem (in L. Nachbin (Ed.), Mathematical Analysis and Applications, Part B, in Advances in Mathematics Supplementary Studies, Vol. 7B, Academic Press, New York, 1981, pp. 741–747), gives a ‘very simple example of a non-Borel separately continuous function $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}^\mathbb{R}$’, but I’ve not seen it. In Borel measurability of separately continuous functions, II, Topology and its Applications 134 (2003), 159–188, he notes that D.K. Burke & R. Pol, On Borel sets in function spaces with the weak topology, J. London Math. Soc.(2) 68 (2003), 725–738, have shown that if $X$ is an infinite compact $F$-space without isolated points or a Baire $P$-space without isolated points, then the evaluation map $e:X\times C_p(X)\to\mathbb{R}:\langle x,f\rangle\mapsto f(x)$ is not Borel measurable, though it is separately continuous. (A space $X$ is an $F$-space if every cozero set in $X$ is $C^*$-embedded in $X$. there are many compact $F$-spaces, since $\beta X\setminus X$ is an $F$-space whenever $X$ is a $\sigma$-compact, locally compact space. A $P$-space is a topological space in which every $G_\delta$-set is open.)