Do multiple network connections between switches increase bandwidth?
Solution 1:
No.
If your unmanaged switches are so simple that they don't do Spanning Tree Protocol (STP), connecting the second cable between them would cause a bridging loop, instantly saturating your network, making your network unusable until you remove the second cable.
If they do do STP, then only one of the switch-to-switch links will be used.
If you had manageable switches instead, then you could configure them to aggregate/trunk multiple ports together, getting a speed increase slightly less than the sum of the aggregated ports' speeds (there is some overhead from aggregation).
Solution 2:
Not unless you configure link aggregation between the switches. Depending on your switch, if it even supports it, this may be referred to as NIC Teaming, Etherchannel, Port Channel, or trunking.
Note that if you simply plug in two connections without configuration such as this, you could seriously degrade your network by causing a loop, as @Spiff mentions in his post.