If you don't use a preposition (e.g. 'on') - i.e. if you say 'I spend time [ing verb]' - what you are saying is that you are 'spending time by [ing verb]'. For example, if I 'spend time reading books', I 'spend time by reading books'. If, however, you do use a preposition and say, 'I spend time on [ing verb]', you are 'using time' to [bare infinitive of ing verb] - so if you 'spend time on reading books', you are 'using time to read books'. Practically, these are saying the same thing, but there are few subtle differences in their thorough meaning. These subtleties, for example, can be exploited in literature texts. For instance, saying, 'he spent months studying physics' (unless in explicit context) is emphasising that he was studying physics out of his own interest and curiosity; 'he spent months on studying physics', on the other hand, may be hinting that he is dedicating some of his free time to studying physics and improving in that area for educational purposes.

The usages that you mention all work slightly differently. I can find no evidence for ‘have time doing something’, so my suspicion is that you cannot infer that particular usage by extension from others. Others are nevertheless interesting.

To spend time doing something

This usually carries the sense of having a finite amount of time available, and allocating some of it to this particular activity.

It can also mean taking care over a task: ‘spending time’ to make sure that something is done well, when one might simply have done a quick job. This would often be slight rephrased as ‘spending time on something’.

Your example ‘I spend too much time watching television’ is a good instance of the first of these.

To have a good time

This means something completely different. Here, ‘time’ means something like ‘experience’, accentuating the sense of a period that you spent doing something. In this usage, the period itself is not the main concern: what you are centrally talking about is the subjective experience. You can ‘have’ a good time, a boring time, a terrifying time, a confusing time, or many other kinds. The word ‘time’ ends up meaning that you can clearly distinguish this experience from what came before and after, and identify it as having had a certain character.

To have time doing something

I have never seen this construction anywhere, so it seems impossible to evaluate your speculation that ‘Have time doing something’ means ‘Spend time doing something’, although the latter is certainly a common expression.

Time to do something

This is similar to ‘spending time’, because it usually relates to the allocation of available time. It might be more common to hear someone say ‘I don’t have time [to do something]’, meaning that it would take too long, so other pressures mean that it cannot be done.

Then again, you can also say that it is time [to do something], meaning that the appropriate moment has arrived: it is time to catch the train, or to change one’s career.

To have time for [something or someone]

This is one that you have not mentioned, but it seems potentially connected. Sometimes this will mean exactly the same as having the time available to do something.

A completely different significance relates to patience or sympathy. If I say that I have time for someone, I am essentially saying that I like or respect them enough to give them my attention or support in some way, i.e. to allocate some of my finite time to them, rather than to something else. Symmetrically, to say ‘I have no time for [someone]’ is to dismiss that person as not being worth spending effort on.

I mention this to help show the range of expressions built on the idea of time as a measurable resource. I can find no evidence for your conjectural ‘have time doing something’, but many related variations certainly exist.

As a crude rule of thumb, if you:

• spend something to do something

... then you spend it before you get or achieve the second thing. Here are some examples:

• We spent a lot of money to ensure the highest quality of workmanship.
• It's time to spend money to create jobs.
• There's no hiding from the fact we spent money to get players in.

In these instances, the infinitives are infinitives of purpose. In other words the infinitive tells us the goal of the spending activity.

In contrast if you:

• spend something doing something

... then you spend it as you are doing it. In other words the spending something and the doing something are happening at the same time:

• We spent a long time cleaning up after the party.
• We spent a lot of energy exercising and fretting over our physical condition.

Notice that in these instances, the -ing-clause tells us how we spent something, not necessarily why we wanted to spend it. Consider the following examples:

• We spent a lot of time getting nowhere
• We spent a lot of time being chased be photographers

Here we did not spend the time with the goal of not making progress or with the goal of being chased.

Gerund participle forms of verbs often indicate simultaneity, in other words they show that two actions are happening at the same time:

• He was run over crossing the road.
• Speaking with his mouth full, he asked me what I'd been doing.

The Original Poster's Question

*We spent a lot of time to shop.

We spent a lot of time shopping.

The Original Poster needs to use the second sentence here. The reason is that the spending time happened concurrently with the shopping. The two activities happened together.