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Let $M$ be Hadamard manifold. Let $p,q\in M$ and $\{p_n\},\{q_n\}\subseteq M$ be such that $p_n\to p$ and $q_n\to q$. In a paper is claimed without proof that $$\exp^{-1}_q{p_n}\to\exp^{-1}_qp,~\exp^{-1}_{p_n}q\to\exp^{-1}_pq~\text{and} \exp^{-1}_{p_n}q_n\to\exp^{-1}_pq$$

For second and third claims we need to the definition of convergence in tangent bundle, while in this paper definition of convergence in tangent bundle is not mentioned. I looked at some books on Riemannian geometry and couldn't find any definition of convergence in tangent bundles, so I think the definition of convergence in tangent bundle should be as follow:

$\{u_n\}\in T_{p_n}M$ converges to $u\in T_pM$ if $\|u_n-P^p_{p_n}(u)\|\to0$,

where $P^p_{p_n}$ is the parallel transport map. I am right? Can anyone refer me to this definition or right definition in some book, etc?

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    To define convergence, you dont need a distance but a topology. Any finite dimensional vector bundle on a manifold is endowed by a topology : Let $x$ a point in this bundle, $x$ its projection on $M$ and i$U\subset M$ an open neighborhood of $p$ over which the bundle is trivial, i.e isomorphic to $U\times R^d$, then a neighborhood of $x$ is just a set which meets $U\times R^d$ in a neighborhood of $p$. This topology is metrizable, of course. – 2017-02-06

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A manifold $M$ is in particular a topological space and you have a notion of convergence of sequences in a topological space. Namely, $x_n \rightarrow x$ is for any open set $U \subseteq M$ containing $x$ there exists $N_U$ such that $x_n \in U$ for all $n > N_U$. This makes sense without introducing any charts but is equivalent to the following condition:

There exists a chart $\varphi \colon U \rightarrow \mathbb{R}^m$ around $x$ and $N \in \mathbb{N}$ such that $x_n \in U$ for $n > N$ and $\varphi(x_n) \rightarrow \varphi(x)$ (where this is the regular convergence of sequences in $\mathbb{R}^m$). Hence, you can check/define the convergence with using charts and this will be independent of the chart used.

All of the above applies of course to $TM$ instead of $M$. Your proposed definition can be made to work but it requires some care because there is no "parallel transport" from $u_n$ to $u$. To get a parallel transport map, you need to specify along which curve you do the parallel transport.