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.

You ask someone else is they can spare the time for something.

Excuse me, could you please spare a few moments of your time?

If you refer to yourself

I can spare some time to talk to you

it makes you sound self-important or condescending.

In the context of your loved ones, again it makes it seem that they are unimportant, if you say you can spare some time to be with them. So

I like to spend time with my loved ones.

You can also ask someone to spend some time with you, if you want to be friendly:

I'd really like it if you could spend some time with me.

If you can spare some time, you make the time available.
If you spend time, you are using the time that you have.

To complicate it slightly further, you might have some spare time that you can spend doing something.

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.