Your post seems to assume something like $\mathsf{P} \neq \mathsf{NP}$ and that an $\mathsf{NP\text{-}complete}$ problem is difficult to solve at least in theory. As you are probably aware, although widely believed by experts, it is unknown.

But let's assume a stronger form of $\mathsf{NP}\neq\mathsf{P}$, for example ETH. SAT is one of the most widely studied $\mathsf {NP\text{-}complete}$ problems (if not the most), and there are easy reductions from SAT to TSP and vice versa. So I will use SAT to answer your questions, similar things apply to TSP (e.g. if you have a small hard instance of SAT then you can convert it to a small hard instance of TSP).

Usually the instances we face in practice have considerably more structure than an arbitrary instance of the problem. Therefore researchers can design algorithms that exploit these structures. There are industrial SAT solver algorithms that are used to solve huge instances of SAT (with several hundred thousands variables) in practice. On the other hand, we know that all these SAT solvers perform exponentially bad on natural, simple, and small formulas like PHP (the pigeon hole principle).

Similar structures exist in TSP problems, e.g. the graph of map of Sweden that you have mentioned is a planar graph, and many graph problems become much simpler when the input is restricted to planar graphs. So if the instances of TSP that you are interested in practice is a subset of all graphs then it is a different problem than the original TSP and it is possible that the restricted problem is not $\mathsf{NP\text{-}hard}$ anymore. (For some graph problems that become easier in restricted cases see this question.)

1. Yes, there are small problem instances for which the algorithms used in practice to solve huge instances fail to solve these small problem instances in reasonable time, e.g. PHP for SAT.

2. It is believed by many that is the case, but we don't have no unconditional exponential time lower-bounds on the running time of (general) algorithms solving these problems.

3. If I understand correctly, you are asking something like: does $TSP \in$ $\mathsf{AveP}$ imply $\mathsf{NP} = \mathsf{P}$? As far as I know, this is unknown.

4. I am not sure what you mean by solving using heuristics methods, but if you mean $\mathsf{HeurP}$, and want to compare it with $\mathsf{P}$, then we should be careful since the problems in these classes are of different types. If you mean the original decision problem plus the uniform distribution, then the Zoo doesn't say much and the answer is probably unknown.

Your post seems to assume something like $\mathsf{P} \neq \mathsf{NP}$ and that an $\mathsf{NP\text{-}complete}$ problem is difficult to solve at least in theory. As you are probably aware, although widely believed by experts, it is unknown.

But let's assume a stronger form of $\mathsf{NP}\neq\mathsf{P}$, for example ETH. SAT is one of the most widely studied $\mathsf {NP\text{-}complete}$ problems (if not the most), and there are easy reductions from SAT to TSP and vice versa. So I will use SAT to answer your questions, similar things apply to TSP (e.g. if you have a small hard instance of SAT then you can convert it to a small hard instance of TSP).

Usually the instances we face in practice have considerably more structure than an arbitrary instance of the problem. Therefore researchers can design algorithms that exploit these structures. There are industrial SAT solver algorithms that are used to solve huge instances of SAT (with several hundred thousands variables) in practice. On the other hand, we know that all these SAT solvers perform exponentially bad on natural, simple, and small formulas like PHP (the pigeon hole principle).

Similar structures exist in TSP problems, e.g. the graph of map of Sweden that you have mentioned is a planar graph, and many graph problems become much simpler when the input is restricted to planar graphs. So if the instances of TSP that you are interested in practice is a subset of all graphs then it is a different problem than the original TSP and it is possible that the restricted problem is not $\mathsf{NP\text{-}hard}$ anymore. (For some graph problems that become easier in restricted cases see this question.)

1. Yes, there are small problem instances for which the algorithms used in practice to solve huge instances fail to solve these small problem instances in reasonable time, e.g. PHP for SAT.

2. It is believed by many that is the case, but we don't have no unconditional exponential time lower-bounds on the running time of (general) algorithms solving these problems.

3. If I understand correctly, you are asking something like: does $TSP \in$ $\mathsf{AveP}$ imply $\mathsf{NP} = \mathsf{P}$? As far as I know, this is unknown.

4. I am not sure what you mean by solving using heuristics methods, but if you mean $\mathsf{HeurP}$, and want to compare it with $\mathsf{P}$, then we should be careful since the problems in these classes are of different types. If you mean the original decision problem plus the uniform distribution, then the Zoo doesn't say much and the answer is probably unknown.

Your post seems to assume something like $\mathsf{P} \neq \mathsf{NP}$ and that an $\mathsf{NP\text{-}complete}$ problem is difficult to solve at least in theory. As you are probably aware, although widely believed by experts, it is unknown.

But let's assume a stronger form of $\mathsf{NP}\neq\mathsf{P}$, for example ETH. SAT is one of the most widely studied $\mathsf {NP\text{-}complete}$ problems (if not the most), and there are easy reductions from SAT to TSP and vice versa. So I will use SAT to answer your questions, similar things apply to TSP (e.g. if you have a small hard instance of SAT then you can convert it to a small hard instance of TSP).

Usually the instances we face in practice have considerably more structure than an arbitrary instance of the problem. Therefore researchers can design algorithms that exploit these structures. There are industrial SAT solver algorithms that are used to solve huge instances of SAT (with several hundred thousands variables) in practice. On the other hand, we know that all these SAT solvers perform exponentially bad on natural, simple, and small formulas like PHP (the pigeon hole principle).

Similar structures exist in TSP problems, e.g. the graph of map of Sweden that you have mentioned is a planner graph, and many graph problems become much simpler when the input is restricted to planner graphs. So if the instances of TSP that you are interested in practice is a subset of all graphs then it is a different problem than the original TSP and it is possible that the restricted problem is not $\mathsf{NP\text{-}hard}$ anymore. (For some graph problems that become easier in restricted cases see this question.)

1. Yes, there are small problem instances for which the algorithms used in practice to solve huge instances fail to solve these small problem instances in reasonable time, e.g. PHP for SAT.

2. It is believed by many that is the case, but we don't have no unconditional exponential time lower-bounds on the running time of (general) algorithms solving these problems.

3. If I understand correctly, you are asking something like: does $TSP \in$ $\mathsf{AveP}$ imply $\mathsf{NP} = \mathsf{P}$? As far as I know, this is unknown.

4. I am not sure what you mean by solving using heuristics methods, but if you mean $\mathsf{HeurP}$, and want to compare it with $\mathsf{P}$, then we should be careful since the problems in these classes are of different types. If you mean the original decision problem plus the uniform distribution, then the Zoo doesn't say much and the answer is probably unknown.

Your post seems to assume something like $\mathsf{P} \neq \mathsf{NP}$ and that an $\mathsf{NP\text{-}complete}$ problem is difficult to solve at least in theory. As you are probably aware, although widely believed by experts, it is unknown.

But let's assume a stronger form of $\mathsf{NP}\neq\mathsf{P}$, for example ETH. SAT is one of the most widely studied $\mathsf {NP\text{-}complete}$ problems (if not the most), and there are easy reductions from SAT to TSP and vice versa. So I will use SAT to answer your questions, similar things apply to TSP (e.g. if you have a small hard instance of SAT then you can convert it to a small hard instance of TSP).

Usually the instances we face in practice have considerably more structure than an arbitrary instance of the problem. Therefore researchers can design algorithms that exploit these structures. There are industrial SAT solver algorithms that are used to solve huge instances of SAT (with several hundred thousands variables) in practice. On the other hand, we know that all these SAT solvers perform exponentially bad on natural, simple, and small formulas like PHP (the pigeon hole principle).

Similar structures exist in TSP problems, e.g. the graph of map of Sweden that you have mentioned is a planar graph, and many graph problems become much simpler when the input is restricted to planar graphs. So if the instances of TSP that you are interested in practice is a subset of all graphs then it is a different problem than the original TSP and it is possible that the restricted problem is not $\mathsf{NP\text{-}hard}$ anymore. (For some graph problems that become easier in restricted cases see this question.)

1. Yes, there are small problem instances for which the algorithms used in practice to solve huge instances fail to solve these small problem instances in reasonable time, e.g. PHP for SAT.

2. It is believed by many that is the case, but we don't have no unconditional exponential time lower-bounds on the running time of (general) algorithms solving these problems.

3. If I understand correctly, you are asking something like: does $TSP \in$ $\mathsf{AveP}$ imply $\mathsf{NP} = \mathsf{P}$? As far as I know, this is unknown.

4. I am not sure what you mean by solving using heuristics methods, but if you mean $\mathsf{HeurP}$, and want to compare it with $\mathsf{P}$, then we should be careful since the problems in these classes are of different types. If you mean the original decision problem plus the uniform distribution, then the Zoo doesn't say much and the answer is probably unknown.